Calculus Examples

$f (x) = \sqrt[3]{| 25 - x^{2} |} + 6$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $\sqrt[3]{| 25 - x^{2} |} + 6$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{| 25 - x^{2} |}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [\sqrt[3]{| 25 - x^{2} |}] + \frac{d}{d x} [6]$

Step 1.2

Evaluate $\frac{d}{d x} [\sqrt[3]{| 25 - x^{2} |}]$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{| 25 - x^{2} |}$ as ${| 25 - x^{2} |}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{| 25 - x^{2} |}^{\frac{1}{3}}] + \frac{d}{d x} [6]$

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = | 25 - x^{2} |$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u_{1}$ as $| 25 - x^{2} |$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{3}}] \frac{d}{d x} [| 25 - x^{2} |] + \frac{d}{d x} [6]$

Step 1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{3}$ .

$\frac{1}{3} {u_{1}}^{\frac{1}{3} - 1} \frac{d}{d x} [| 25 - x^{2} |] + \frac{d}{d x} [6]$

Step 1.2.2.3

Replace all occurrences of $u_{1}$ with $| 25 - x^{2} |$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} \frac{d}{d x} [| 25 - x^{2} |] + \frac{d}{d x} [6]$

Step 1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 25 - x^{2}$ .

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Step 1.2.3.1

To apply the Chain Rule, set $u_{2}$ as $25 - x^{2}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [25 - x^{2}]) + \frac{d}{d x} [6]$

Step 1.2.3.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [25 - x^{2}]) + \frac{d}{d x} [6]$

Step 1.2.3.3

Replace all occurrences of $u_{2}$ with $25 - x^{2}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} \frac{d}{d x} [25 - x^{2}]) + \frac{d}{d x} [6]$

Step 1.2.4

By the Sum Rule, the derivative of $25 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}])) + \frac{d}{d x} [6]$

Step 1.2.5

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 + \frac{d}{d x} [- x^{2}])) + \frac{d}{d x} [6]$

Step 1.2.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - \frac{d}{d x} [x^{2}])) + \frac{d}{d x} [6]$

Step 1.2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 1.2.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 1.2.9

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 1.2.10

Combine the numerators over the common denominator.

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1 - 1 \cdot 3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 1.2.11

Simplify the numerator.

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Step 1.2.11.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1 - 3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 1.2.11.2

Subtract $3$ from $1$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{- 2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 1.2.12

Move the negative in front of the fraction.

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 1.2.13

Multiply $2$ by $- 1$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - 2 x)) + \frac{d}{d x} [6]$

Step 1.2.14

Subtract $2 x$ from $0$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (- 2 x)) + \frac{d}{d x} [6]$

Step 1.2.15

Combine $- 2$ and $\frac{25 - x^{2}}{| 25 - x^{2} |}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{- 2 (25 - x^{2})}{| 25 - x^{2} |} x) + \frac{d}{d x} [6]$

Step 1.2.16

Combine $\frac{- 2 (25 - x^{2})}{| 25 - x^{2} |}$ and $x$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} \frac{- 2 (25 - x^{2}) x}{| 25 - x^{2} |} + \frac{d}{d x} [6]$

Step 1.2.17

Move the negative in front of the fraction.

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (- \frac{(2 (25 - x^{2})) x}{| 25 - x^{2} |}) + \frac{d}{d x} [6]$

Step 1.2.18

Combine $\frac{1}{3}$ and ${| 25 - x^{2} |}^{- \frac{2}{3}}$ .

$\frac{{| 25 - x^{2} |}^{- \frac{2}{3}}}{3} (- \frac{(2 (25 - x^{2})) x}{| 25 - x^{2} |}) + \frac{d}{d x} [6]$

Step 1.2.19

Multiply $\frac{{| 25 - x^{2} |}^{- \frac{2}{3}}}{3}$ by $\frac{(2 (25 - x^{2})) x}{| 25 - x^{2} |}$ .

$- \frac{{| 25 - x^{2} |}^{- \frac{2}{3}} ((2 (25 - x^{2})) x)}{3 | 25 - x^{2} |} + \frac{d}{d x} [6]$

Step 1.2.20

Move ${| 25 - x^{2} |}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{(2 (25 - x^{2})) x}{3 | 25 - x^{2} | {| 25 - x^{2} |}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 1.2.21

Multiply $| 25 - x^{2} |$ by ${| 25 - x^{2} |}^{\frac{2}{3}}$ by adding the exponents.

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Step 1.2.21.1

Move ${| 25 - x^{2} |}^{\frac{2}{3}}$ .

$- \frac{(2 (25 - x^{2})) x}{3 ({| 25 - x^{2} |}^{\frac{2}{3}} | 25 - x^{2} |)} + \frac{d}{d x} [6]$

Step 1.2.21.2

Multiply ${| 25 - x^{2} |}^{\frac{2}{3}}$ by $| 25 - x^{2} |$ .

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Step 1.2.21.2.1

Raise $| 25 - x^{2} |$ to the power of $1$ .

$- \frac{(2 (25 - x^{2})) x}{3 ({| 25 - x^{2} |}^{\frac{2}{3}} {| 25 - x^{2} |}^{1})} + \frac{d}{d x} [6]$

Step 1.2.21.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{2}{3} + 1}} + \frac{d}{d x} [6]$

Step 1.2.21.3

Write $1$ as a fraction with a common denominator.

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{2}{3} + \frac{3}{3}}} + \frac{d}{d x} [6]$

Step 1.2.21.4

Combine the numerators over the common denominator.

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{2 + 3}{3}}} + \frac{d}{d x} [6]$

Step 1.2.21.5

Add $2$ and $3$ .

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + \frac{d}{d x} [6]$

Step 1.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$- \frac{(2 \cdot 25 + 2 (- x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4.2

Apply the distributive property.

$- \frac{2 \cdot 25 x + 2 (- x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4.3

Combine terms.

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Step 1.4.3.1

Multiply $2$ by $25$ .

$- \frac{50 x + 2 (- x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4.3.2

Multiply $- 1$ by $2$ .

$- \frac{50 x - 2 x^{2} x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4.3.3

Raise $x$ to the power of $1$ .

$- \frac{50 x - 2 (x^{1} x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{50 x - 2 x^{1 + 2}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4.3.5

Add $1$ and $2$ .

$- \frac{50 x - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 1.4.3.6

Add $- \frac{50 x - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ and $0$ .

$- \frac{50 x - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 1.4.4

Simplify the numerator.

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Step 1.4.4.1

Factor $2 x$ out of $50 x - 2 x^{3}$ .

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Step 1.4.4.1.1

Factor $2 x$ out of $50 x$ .

$- \frac{2 x (25) - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 1.4.4.1.2

Factor $2 x$ out of $- 2 x^{3}$ .

$- \frac{2 x (25) + 2 x (- x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 1.4.4.1.3

Factor $2 x$ out of $2 x (25) + 2 x (- x^{2})$ .

$- \frac{2 x (25 - x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 1.4.4.2

Rewrite $25$ as $5^{2}$ .

$- \frac{2 x (5^{2} - x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 1.4.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ with respect to $x$ is $- \frac{2}{3} \frac{d}{d x} [\frac{(x (5 + x)) (5 - x)}{{| 25 - x^{2} |}^{\frac{5}{3}}}]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{d}{d x} \frac{(x (5 + x)) (5 - x)}{{| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x (5 + x) (5 - x)$ and $g (x) = {| 25 - x^{2} |}^{\frac{5}{3}}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} \frac{d}{d x} (x (5 + x) (5 - x)) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{({| 25 - x^{2} |}^{\frac{5}{3}})}^{2}}$

Step 2.3

Multiply the exponents in ${({| 25 - x^{2} |}^{\frac{5}{3}})}^{2}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} \frac{d}{d x} (x (5 + x) (5 - x)) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{5}{3} \cdot 2}}$

Step 2.3.2

Multiply $\frac{5}{3} \cdot 2$ .

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Step 2.3.2.1

Combine $\frac{5}{3}$ and $2$ .

Step 2.3.2.2

Multiply $5$ by $2$ .

Step 2.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x (5 + x)$ and $g (x) = 5 - x$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (x (5 + x) \frac{d}{d x} (5 - x) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $5 - x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (x (5 + x) (\frac{d}{d x} (5) + \frac{d}{d x} (- x)) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (x (5 + x) (0 + \frac{d}{d x} (- x)) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (x (5 + x) \frac{d}{d x} (- x) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (x (5 + x) (- \frac{d}{d x} x) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (x (5 + x) (- 1 \cdot 1) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5.6

Simplify the expression.

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Step 2.5.6.1

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (x (5 + x) \cdot - 1 + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5.6.2

Move $- 1$ to the left of $x (5 + x)$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- 1 \cdot (x (5 + x)) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.5.6.3

Rewrite $- 1 x$ as $- x$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) \frac{d}{d x} (x (5 + x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 5 + x$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x \frac{d}{d x} (5 + x) + (5 + x) \frac{d}{d x} (x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7

Differentiate.

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Step 2.7.1

By the Sum Rule, the derivative of $5 + x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x (\frac{d}{d x} (5) + \frac{d}{d x} (x)) + (5 + x) \frac{d}{d x} (x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x (0 + \frac{d}{d x} (x)) + (5 + x) \frac{d}{d x} (x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7.3

Add $0$ and $\frac{d}{d x} [x]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x \frac{d}{d x} (x) + (5 + x) \frac{d}{d x} (x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x \cdot 1 + (5 + x) \frac{d}{d x} (x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7.5

Multiply $x$ by $1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x + (5 + x) \frac{d}{d x} (x))) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x + (5 + x) \cdot 1)) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7.7

Simplify by adding terms.

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Step 2.7.7.1

Multiply $5 + x$ by $1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (x + 5 + x)) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.7.7.2

Add $x$ and $x$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) \frac{d}{d x} {| 25 - x^{2} |}^{\frac{5}{3}}}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{5}{3}}$ and $g (x) = | 25 - x^{2} |$ .

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Step 2.8.1

To apply the Chain Rule, set $u_{1}$ as $| 25 - x^{2} |$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{d}{d u (1)} ({u_{1}}^{\frac{5}{3}}) \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.8.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{5}{3}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5}{3} {u_{1}}^{\frac{5}{3} - 1} \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.8.3

Replace all occurrences of $u_{1}$ with $| 25 - x^{2} |$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5}{3} \cdot {| 25 - x^{2} |}^{\frac{5}{3} - 1} \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5}{3} \cdot {| 25 - x^{2} |}^{\frac{5}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.10

Combine $- 1$ and $\frac{3}{3}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5}{3} \cdot {| 25 - x^{2} |}^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.11

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5}{3} \cdot {| 25 - x^{2} |}^{\frac{5 - 1 \cdot 3}{3}} \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.12

Simplify the numerator.

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Step 2.12.1

Multiply $- 1$ by $3$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5}{3} \cdot {| 25 - x^{2} |}^{\frac{5 - 3}{3}} \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.12.2

Subtract $3$ from $5$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5}{3} \cdot {| 25 - x^{2} |}^{\frac{2}{3}} \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.13

Combine fractions.

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Step 2.13.1

Combine $\frac{5}{3}$ and ${| 25 - x^{2} |}^{\frac{2}{3}}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - x (5 + x) (5 - x) (\frac{5 {| 25 - x^{2} |}^{\frac{2}{3}}}{3} \cdot \frac{d}{d x} (| 25 - x^{2} |))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.13.2

Combine $\frac{5 {| 25 - x^{2} |}^{\frac{2}{3}}}{3}$ and $x$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 {| 25 - x^{2} |}^{\frac{2}{3}} x}{3} \cdot ((5 + x) (5 - x)) \frac{d}{d x} | 25 - x^{2} |}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.14

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 25 - x^{2}$ .

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Step 2.14.1

To apply the Chain Rule, set $u_{2}$ as $25 - x^{2}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 {| 25 - x^{2} |}^{\frac{2}{3}} x}{3} \cdot ((5 + x) (5 - x) (\frac{d}{d u (2)} (| u_{2} |) \frac{d}{d x} (25 - x^{2})))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.14.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 {| 25 - x^{2} |}^{\frac{2}{3}} x}{3} \cdot ((5 + x) (5 - x) (\frac{u_{2}}{| u_{2} |} \cdot \frac{d}{d x} (25 - x^{2})))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.14.3

Replace all occurrences of $u_{2}$ with $25 - x^{2}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 {| 25 - x^{2} |}^{\frac{2}{3}} x}{3} \cdot ((5 + x) (5 - x) (\frac{25 - x^{2}}{| 25 - x^{2} |} \cdot \frac{d}{d x} (25 - x^{2})))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.15

Combine fractions.

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Step 2.15.1

Multiply $\frac{25 - x^{2}}{| 25 - x^{2} |}$ by $\frac{5 {| 25 - x^{2} |}^{\frac{2}{3}} x}{3}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{(25 - x^{2}) (5 {| 25 - x^{2} |}^{\frac{2}{3}} x)}{| 25 - x^{2} | \cdot 3} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.15.2

Simplify the expression.

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Step 2.15.2.1

Move $5$ to the left of $25 - x^{2}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 \cdot ((25 - x^{2}) {| 25 - x^{2} |}^{\frac{2}{3}} x)}{| 25 - x^{2} | \cdot 3} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.15.2.2

Move $3$ to the left of $| 25 - x^{2} |$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) {| 25 - x^{2} |}^{\frac{2}{3}} x}{3 \cdot | 25 - x^{2} |} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.15.2.3

Move ${| 25 - x^{2} |}^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 | 25 - x^{2} | {| 25 - x^{2} |}^{- \frac{2}{3}}} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.16

Multiply $| 25 - x^{2} |$ by ${| 25 - x^{2} |}^{- \frac{2}{3}}$ by adding the exponents.

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Step 2.16.1

Move ${| 25 - x^{2} |}^{- \frac{2}{3}}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 ({| 25 - x^{2} |}^{- \frac{2}{3}} | 25 - x^{2} |)} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.16.2

Multiply ${| 25 - x^{2} |}^{- \frac{2}{3}}$ by $| 25 - x^{2} |$ .

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Step 2.16.2.1

Raise $| 25 - x^{2} |$ to the power of $1$ .

Step 2.16.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{- \frac{2}{3} + 1}} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.16.3

Write $1$ as a fraction with a common denominator.

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{- \frac{2}{3} + \frac{3}{3}}} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.16.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{- 2 + 3}{3}}} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.16.5

Add $- 2$ and $3$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (25 - x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.17

By the Sum Rule, the derivative of $25 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x) (\frac{d}{d x} (25) + \frac{d}{d x} (- x^{2})))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.18

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x) (0 + \frac{d}{d x} (- x^{2})))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.19

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

Step 2.20

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) - \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x) (- \frac{d}{d x} x^{2}))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.21

Multiply.

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Step 2.21.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + 1 (\frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}) \cdot ((5 + x) (5 - x)) \frac{d}{d x} (x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.21.2

Multiply $\frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ by $1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)) \frac{d}{d x} (x^{2})}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.22

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x) (2 x))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.23

Combine fractions.

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Step 2.23.1

Combine $2$ and $\frac{5 (25 - x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{2 (5 (25 - x^{2}) x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x) x)}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.23.2

Multiply $5$ by $2$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{10 ((25 - x^{2}) x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x) x)}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.23.3

Combine $x$ and $\frac{10 ((25 - x^{2}) x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x (10 ((25 - x^{2}) x))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.24

Raise $x$ to the power of $1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x \cdot x (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.25

Raise $x$ to the power of $1$ .

Step 2.26

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x^{1 + 1} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.27

Add $1$ and $1$ .

$f'' (x) = - \frac{2}{3} \cdot \frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x^{2} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x))}{{| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.28

Multiply $\frac{{| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x^{2} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} (5 + x) (5 - x)}{{| 25 - x^{2} |}^{\frac{10}{3}}}$ by $\frac{2}{3}$ .

$f'' (x) = - \frac{({| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x^{2} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x))) \cdot 2}{{| 25 - x^{2} |}^{\frac{10}{3}} \cdot 3}$

Step 2.29

Reorder.

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Step 2.29.1

Move $2$ to the left of ${| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x^{2} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} (5 + x) (5 - x)$ .

$f'' (x) = - \frac{2 \cdot ({| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x^{2} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{{| 25 - x^{2} |}^{\frac{10}{3}} \cdot 3}$

Step 2.29.2

Move $3$ to the left of ${| 25 - x^{2} |}^{\frac{10}{3}}$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- x (5 + x) + (5 - x) (2 x + 5)) + \frac{x^{2} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 \cdot {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30

Simplify.

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Step 2.30.1

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- x \cdot 5 - x \cdot x + (5 - x) (2 x + 5)) + \frac{x^{2} (10 (25 - x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.2

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- x \cdot 5 - x \cdot x + (5 - x) (2 x + 5)) + \frac{x^{2} (10 \cdot 25 + 10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.3

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- x \cdot 5 - x \cdot x + (5 - x) (2 x + 5)) + \frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.4

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- x \cdot 5 - x \cdot x + (5 - x) (2 x + 5))) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5

Simplify the numerator.

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Step 2.30.5.1

Simplify each term.

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Step 2.30.5.1.1

Simplify each term.

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Step 2.30.5.1.1.1

Multiply $5$ by $- 1$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x \cdot x + (5 - x) (2 x + 5))) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.30.5.1.1.2.1

Move $x$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - (x \cdot x) + (5 - x) (2 x + 5))) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + (5 - x) (2 x + 5))) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.3

Expand $(5 - x) (2 x + 5)$ using the FOIL Method.

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Step 2.30.5.1.1.3.1

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 5 (2 x + 5) - x (2 x + 5))) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.3.2

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 5 (2 x) + 5 \cdot 5 - x (2 x + 5))) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.3.3

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 5 (2 x) + 5 \cdot 5 - x (2 x) - x \cdot 5)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4

Simplify and combine like terms.

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Step 2.30.5.1.1.4.1

Simplify each term.

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Step 2.30.5.1.1.4.1.1

Multiply $2$ by $5$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 10 x + 5 \cdot 5 - x (2 x) - x \cdot 5)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4.1.2

Multiply $5$ by $5$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 10 x + 25 - x (2 x) - x \cdot 5)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4.1.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 10 x + 25 - 1 \cdot (2 x \cdot x) - x \cdot 5)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.30.5.1.1.4.1.4.1

Move $x$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 10 x + 25 - 1 \cdot (2 (x \cdot x)) - x \cdot 5)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4.1.4.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 10 x + 25 - 1 \cdot (2 x^{2}) - x \cdot 5)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4.1.5

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 10 x + 25 - 2 x^{2} - x \cdot 5)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4.1.6

Multiply $5$ by $- 1$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 10 x + 25 - 2 x^{2} - 5 x)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.1.4.2

Subtract $5 x$ from $10 x$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 5 x - x^{2} + 5 x + 25 - 2 x^{2})) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.2

Combine the opposite terms in $- 5 x - x^{2} + 5 x + 25 - 2 x^{2}$ .

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Step 2.30.5.1.2.1

Add $- 5 x$ and $5 x$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- x^{2} + 0 + 25 - 2 x^{2})) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.2.2

Add $- x^{2}$ and $0$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- x^{2} + 25 - 2 x^{2})) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.3

Subtract $2 x^{2}$ from $- x^{2}$ .

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 3 x^{2} + 25)) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.4

Apply the distributive property.

$f'' (x) = - \frac{2 ({| 25 - x^{2} |}^{\frac{5}{3}} (- 3 x^{2}) + {| 25 - x^{2} |}^{\frac{5}{3}} \cdot 25) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.5

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{2 (- 3 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + {| 25 - x^{2} |}^{\frac{5}{3}} \cdot 25) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.6

Move $25$ to the left of ${| 25 - x^{2} |}^{\frac{5}{3}}$ .

$f'' (x) = - \frac{2 (- 3 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 25 \cdot {| 25 - x^{2} |}^{\frac{5}{3}}) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.7

Apply the distributive property.

$f'' (x) = - \frac{2 (- 3 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2}) + 2 (25 {| 25 - x^{2} |}^{\frac{5}{3}}) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.8

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{- 6 ({| 25 - x^{2} |}^{\frac{5}{3}} x^{2}) + 2 (25 {| 25 - x^{2} |}^{\frac{5}{3}}) + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.9

Multiply $25$ by $2$ .

$f'' (x) = - \frac{- 6 ({| 25 - x^{2} |}^{\frac{5}{3}} x^{2}) + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{x^{2} (10 \cdot 25) + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10

Simplify the numerator.

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Step 2.30.5.1.10.1

Multiply $10$ by $25$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{x^{2} \cdot 250 + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.2

Move $250$ to the left of $x^{2}$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{250 \cdot x^{2} + x^{2} (10 (- x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{250 x^{2} + 10 \cdot (- 1 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.1.10.4.1

Move $x^{2}$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{250 x^{2} + 10 \cdot (- 1 (x^{2} x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{250 x^{2} + 10 \cdot (- 1 x^{2 + 2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.4.3

Add $2$ and $2$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{250 x^{2} + 10 \cdot (- 1 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.5

Multiply $10$ by $- 1$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{250 x^{2} - 10 x^{4}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.6

Rewrite $250 x^{2} - 10 x^{4}$ in a factored form.

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Step 2.30.5.1.10.6.1

Factor $10 x^{2}$ out of $250 x^{2} - 10 x^{4}$ .

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Step 2.30.5.1.10.6.1.1

Factor $10 x^{2}$ out of $250 x^{2}$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} (25) - 10 x^{4}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.6.1.2

Factor $10 x^{2}$ out of $- 10 x^{4}$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} (25) + 10 x^{2} (- x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.6.1.3

Factor $10 x^{2}$ out of $10 x^{2} (25) + 10 x^{2} (- x^{2})$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} (25 - x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.6.2

Rewrite $25$ as $5^{2}$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} (5^{2} - x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.10.6.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot ((5 + x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.11

Multiply $\frac{10 x^{2} (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ by $5 + x$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} (5 + x) (5 - x) (5 + x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot (5 - x))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.12

Simplify the numerator.

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Step 2.30.5.1.12.1

Raise $5 + x$ to the power of $1$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} ((5 + x) (5 + x)) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot (5 - x))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.12.2

Raise $5 + x$ to the power of $1$ .

Step 2.30.5.1.12.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} {(5 + x)}^{1 + 1} (5 - x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot (5 - x))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.12.4

Add $1$ and $1$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} {(5 + x)}^{2} (5 - x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot (5 - x))}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.13

Multiply $\frac{10 x^{2} {(5 + x)}^{2} (5 - x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ by $5 - x$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} {(5 + x)}^{2} (5 - x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}})}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.14

Simplify the numerator.

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Step 2.30.5.1.14.1

Raise $5 - x$ to the power of $1$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} {(5 + x)}^{2} ((5 - x) (5 - x))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}})}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.14.2

Raise $5 - x$ to the power of $1$ .

Step 2.30.5.1.14.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} {(5 + x)}^{2} {(5 - x)}^{1 + 1}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}})}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.14.4

Add $1$ and $1$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + 2 (\frac{10 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}})}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.15

Multiply $2 \frac{10 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

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Step 2.30.5.1.15.1

Combine $2$ and $\frac{10 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 (10 x^{2} {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.1.15.2

Multiply $10$ by $2$ .

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{20 (x^{2} {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

$f'' (x) = - \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} + 50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{20 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.2

To write $- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2}$ as a fraction with a common denominator, multiply by $\frac{3 {| 25 - x^{2} |}^{\frac{1}{3}}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} - 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} \cdot \frac{3 {| 25 - x^{2} |}^{\frac{1}{3}}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} + \frac{20 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.3

Combine $- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2}$ and $\frac{3 {| 25 - x^{2} |}^{\frac{1}{3}}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} (3 {| 25 - x^{2} |}^{\frac{1}{3}})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} + \frac{20 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} (3 {| 25 - x^{2} |}^{\frac{1}{3}}) + 20 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5

Simplify the numerator.

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Step 2.30.5.5.1

Factor $2 x^{2}$ out of $- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} (3 {| 25 - x^{2} |}^{\frac{1}{3}}) + 20 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}$ .

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Step 2.30.5.5.1.1

Factor $2 x^{2}$ out of $- 6 {| 25 - x^{2} |}^{\frac{5}{3}} x^{2} (3 {| 25 - x^{2} |}^{\frac{1}{3}})$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 3 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})) + 20 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.1.2

Factor $2 x^{2}$ out of $20 x^{2} {(5 + x)}^{2} {(5 - x)}^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 3 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})) + 2 x^{2} (10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.1.3

Factor $2 x^{2}$ out of $2 x^{2} (- 3 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})) + 2 x^{2} (10 {(5 + x)}^{2} {(5 - x)}^{2})$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 3 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}}) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.2

Combine exponents.

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Step 2.30.5.5.2.1

Multiply $3$ by $- 3$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 {| 25 - x^{2} |}^{\frac{5}{3}} {| 25 - x^{2} |}^{\frac{1}{3}} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.2.2

Multiply ${| 25 - x^{2} |}^{\frac{5}{3}}$ by ${| 25 - x^{2} |}^{\frac{1}{3}}$ by adding the exponents.

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Step 2.30.5.5.2.2.1

Move ${| 25 - x^{2} |}^{\frac{1}{3}}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 ({| 25 - x^{2} |}^{\frac{1}{3}} {| 25 - x^{2} |}^{\frac{5}{3}}) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 {| 25 - x^{2} |}^{\frac{1}{3} + \frac{5}{3}} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.2.2.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 {| 25 - x^{2} |}^{\frac{1 + 5}{3}} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.2.2.4

Add $1$ and $5$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 {| 25 - x^{2} |}^{\frac{6}{3}} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.2.2.5

Divide $6$ by $3$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 {| 25 - x^{2} |}^{2} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3

Simplify each term.

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Step 2.30.5.5.3.1

Remove the absolute value in ${| 25 - x^{2} |}^{2}$ because exponentiations with even powers are always positive.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 {(25 - x^{2})}^{2} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.2

Rewrite ${(25 - x^{2})}^{2}$ as $(25 - x^{2}) (25 - x^{2})$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 ((25 - x^{2}) (25 - x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.3

Expand $(25 - x^{2}) (25 - x^{2})$ using the FOIL Method.

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Step 2.30.5.5.3.3.1

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (25 (25 - x^{2}) - x^{2} (25 - x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.3.2

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (25 \cdot 25 + 25 (- x^{2}) - x^{2} (25 - x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.3.3

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (25 \cdot 25 + 25 (- x^{2}) - x^{2} \cdot 25 - x^{2} (- x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4

Simplify and combine like terms.

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Step 2.30.5.5.3.4.1

Simplify each term.

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Step 2.30.5.5.3.4.1.1

Multiply $25$ by $25$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 + 25 (- x^{2}) - x^{2} \cdot 25 - x^{2} (- x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.2

Multiply $- 1$ by $25$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - x^{2} \cdot 25 - x^{2} (- x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.3

Multiply $25$ by $- 1$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - 25 x^{2} - x^{2} (- x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 x^{2} x^{2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.5.3.4.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 (x^{2} x^{2}))) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 x^{2 + 2})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 x^{4})) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - 25 x^{2} + 1 x^{4}) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.1.7

Multiply $x^{4}$ by $1$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 25 x^{2} - 25 x^{2} + x^{4}) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.4.2

Subtract $25 x^{2}$ from $- 25 x^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 (625 - 50 x^{2} + x^{4}) + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.5

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 9 \cdot 625 - 9 (- 50 x^{2}) - 9 x^{4} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.6

Simplify.

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Step 2.30.5.5.3.6.1

Multiply $- 9$ by $625$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 - 9 (- 50 x^{2}) - 9 x^{4} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.6.2

Multiply $- 50$ by $- 9$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 {(5 + x)}^{2} {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.7

Rewrite ${(5 + x)}^{2}$ as $(5 + x) (5 + x)$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 ((5 + x) (5 + x)) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.8

Expand $(5 + x) (5 + x)$ using the FOIL Method.

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Step 2.30.5.5.3.8.1

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 (5 (5 + x) + x (5 + x)) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.8.2

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 (5 \cdot 5 + 5 x + x (5 + x)) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.8.3

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 (5 \cdot 5 + 5 x + x \cdot 5 + x \cdot x) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.9

Simplify and combine like terms.

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Step 2.30.5.5.3.9.1

Simplify each term.

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Step 2.30.5.5.3.9.1.1

Multiply $5$ by $5$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 (25 + 5 x + x \cdot 5 + x \cdot x) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.9.1.2

Move $5$ to the left of $x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 (25 + 5 x + 5 \cdot x + x \cdot x) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.9.1.3

Multiply $x$ by $x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 (25 + 5 x + 5 x + x^{2}) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.9.2

Add $5 x$ and $5 x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 10 (25 + 10 x + x^{2}) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.10

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (10 \cdot 25 + 10 (10 x) + 10 x^{2}) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.11

Simplify.

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Step 2.30.5.5.3.11.1

Multiply $10$ by $25$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 10 (10 x) + 10 x^{2}) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.11.2

Multiply $10$ by $10$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) {(5 - x)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.12

Rewrite ${(5 - x)}^{2}$ as $(5 - x) (5 - x)$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) ((5 - x) (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.13

Expand $(5 - x) (5 - x)$ using the FOIL Method.

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Step 2.30.5.5.3.13.1

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (5 (5 - x) - x (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.13.2

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (5 \cdot 5 + 5 (- x) - x (5 - x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.13.3

Apply the distributive property.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (5 \cdot 5 + 5 (- x) - x \cdot 5 - x (- x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14

Simplify and combine like terms.

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Step 2.30.5.5.3.14.1

Simplify each term.

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Step 2.30.5.5.3.14.1.1

Multiply $5$ by $5$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 + 5 (- x) - x \cdot 5 - x (- x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.1.2

Multiply $- 1$ by $5$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 5 x - x \cdot 5 - x (- x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.1.3

Multiply $5$ by $- 1$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 5 x - 5 x - x (- x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 5 x - 5 x - 1 \cdot (- 1 x \cdot x)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.30.5.5.3.14.1.5.1

Move $x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 5 x - 5 x - 1 \cdot (- 1 (x \cdot x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.1.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 5 x - 5 x - 1 \cdot (- 1 x^{2})))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 5 x - 5 x + 1 x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.1.7

Multiply $x^{2}$ by $1$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 5 x - 5 x + x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.14.2

Subtract $5 x$ from $- 5 x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + (250 + 100 x + 10 x^{2}) (25 - 10 x + x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.15

Expand $(250 + 100 x + 10 x^{2}) (25 - 10 x + x^{2})$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 250 \cdot 25 + 250 (- 10 x) + 250 x^{2} + 100 x \cdot 25 + 100 x (- 10 x) + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16

Simplify each term.

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Step 2.30.5.5.3.16.1

Multiply $250$ by $25$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 + 250 (- 10 x) + 250 x^{2} + 100 x \cdot 25 + 100 x (- 10 x) + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.2

Multiply $- 10$ by $250$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 100 x \cdot 25 + 100 x (- 10 x) + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.3

Multiply $25$ by $100$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x + 100 x (- 10 x) + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x + 100 \cdot (- 10 x \cdot x) + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.30.5.5.3.16.5.1

Move $x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x + 100 \cdot (- 10 (x \cdot x)) + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x + 100 \cdot (- 10 x^{2}) + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.6

Multiply $100$ by $- 10$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x \cdot x^{2} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.7

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.5.3.16.7.1

Move $x^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 (x^{2} x) + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.7.2

Multiply $x^{2}$ by $x$ .

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Step 2.30.5.5.3.16.7.2.1

Raise $x$ to the power of $1$ .

Step 2.30.5.5.3.16.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{2 + 1} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.7.3

Add $2$ and $1$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 10 x^{2} \cdot 25 + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.8

Multiply $25$ by $10$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} + 10 x^{2} (- 10 x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.9

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} + 10 \cdot (- 10 x^{2} x) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.10

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.30.5.5.3.16.10.1

Move $x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} + 10 \cdot (- 10 (x \cdot x^{2})) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.10.2

Multiply $x$ by $x^{2}$ .

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Step 2.30.5.5.3.16.10.2.1

Raise $x$ to the power of $1$ .

Step 2.30.5.5.3.16.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} + 10 \cdot (- 10 x^{1 + 2}) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.10.3

Add $1$ and $2$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} + 10 \cdot (- 10 x^{3}) + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.11

Multiply $10$ by $- 10$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} - 100 x^{3} + 10 x^{2} x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.12

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.5.3.16.12.1

Move $x^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} - 100 x^{3} + 10 (x^{2} x^{2}))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.12.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} - 100 x^{3} + 10 x^{2 + 2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.16.12.3

Add $2$ and $2$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} - 100 x^{3} + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.17

Combine the opposite terms in $6250 - 2500 x + 250 x^{2} + 2500 x - 1000 x^{2} + 100 x^{3} + 250 x^{2} - 100 x^{3} + 10 x^{4}$ .

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Step 2.30.5.5.3.17.1

Add $- 2500 x$ and $2500 x$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 + 250 x^{2} + 0 - 1000 x^{2} + 100 x^{3} + 250 x^{2} - 100 x^{3} + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.17.2

Add $6250 + 250 x^{2}$ and $0$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 + 250 x^{2} - 1000 x^{2} + 100 x^{3} + 250 x^{2} - 100 x^{3} + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.17.3

Subtract $100 x^{3}$ from $100 x^{3}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 + 250 x^{2} - 1000 x^{2} + 250 x^{2} + 0 + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.17.4

Add $6250 + 250 x^{2} - 1000 x^{2} + 250 x^{2}$ and $0$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 + 250 x^{2} - 1000 x^{2} + 250 x^{2} + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.18

Subtract $1000 x^{2}$ from $250 x^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 750 x^{2} + 250 x^{2} + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.3.19

Add $- 750 x^{2}$ and $250 x^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 5625 + 450 x^{2} - 9 x^{4} + 6250 - 500 x^{2} + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.4

Add $- 5625$ and $6250$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (450 x^{2} - 9 x^{4} + 625 - 500 x^{2} + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.5

Subtract $500 x^{2}$ from $450 x^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 50 x^{2} - 9 x^{4} + 625 + 10 x^{4})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.6

Add $- 9 x^{4}$ and $10 x^{4}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (- 50 x^{2} + x^{4} + 625)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.7

Factor using the perfect square rule.

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Step 2.30.5.5.7.1

Rearrange terms.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} (x^{4} - 50 x^{2} + 625)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.7.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} ({(x^{2})}^{2} - 50 x^{2} + 625)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.7.3

Rewrite $625$ as $25^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} ({(x^{2})}^{2} - 50 x^{2} + 25^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.7.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$50 x^{2} = 2 \cdot x^{2} \cdot 25$

Step 2.30.5.5.7.5

Rewrite the polynomial.

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} ({(x^{2})}^{2} - 2 \cdot x^{2} \cdot 25 + 25^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.7.6

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x^{2}$ and $b = 25$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} {(x^{2} - 25)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.8

Rewrite $25$ as $5^{2}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} {(x^{2} - 5^{2})}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.9

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} {((x + 5) (x - 5))}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.5.10

Apply the product rule to $(x + 5) (x - 5)$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} + \frac{2 x^{2} {(x + 5)}^{2} {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.6

To write $50 {| 25 - x^{2} |}^{\frac{5}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {| 25 - x^{2} |}^{\frac{1}{3}}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} \cdot \frac{3 {| 25 - x^{2} |}^{\frac{1}{3}}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} + \frac{2 x^{2} {(x + 5)}^{2} {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.7

Combine $50 {| 25 - x^{2} |}^{\frac{5}{3}}$ and $\frac{3 {| 25 - x^{2} |}^{\frac{1}{3}}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ .

$f'' (x) = - \frac{\frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} + \frac{2 x^{2} {(x + 5)}^{2} {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.8

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{50 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}}) + 2 x^{2} {(x + 5)}^{2} {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9

Simplify the numerator.

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Step 2.30.5.9.1

Factor $2$ out of $50 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}}) + 2 x^{2} {(x + 5)}^{2} {(x - 5)}^{2}$ .

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Step 2.30.5.9.1.1

Factor $2$ out of $50 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})$ .

$f'' (x) = - \frac{\frac{2 (25 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})) + 2 x^{2} {(x + 5)}^{2} {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.1.2

Factor $2$ out of $2 x^{2} {(x + 5)}^{2} {(x - 5)}^{2}$ .

$f'' (x) = - \frac{\frac{2 (25 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})) + 2 (x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.1.3

Factor $2$ out of $2 (25 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}})) + 2 (x^{2} {(x + 5)}^{2} {(x - 5)}^{2})$ .

$f'' (x) = - \frac{\frac{2 (25 {| 25 - x^{2} |}^{\frac{5}{3}} (3 {| 25 - x^{2} |}^{\frac{1}{3}}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.2

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (25 \cdot (3 {| 25 - x^{2} |}^{\frac{5}{3}} {| 25 - x^{2} |}^{\frac{1}{3}}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.3

Multiply ${| 25 - x^{2} |}^{\frac{5}{3}}$ by ${| 25 - x^{2} |}^{\frac{1}{3}}$ by adding the exponents.

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Step 2.30.5.9.3.1

Move ${| 25 - x^{2} |}^{\frac{1}{3}}$ .

$f'' (x) = - \frac{\frac{2 (25 \cdot (3 ({| 25 - x^{2} |}^{\frac{1}{3}} {| 25 - x^{2} |}^{\frac{5}{3}})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (25 \cdot (3 {| 25 - x^{2} |}^{\frac{1}{3} + \frac{5}{3}}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.3.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{2 (25 \cdot (3 {| 25 - x^{2} |}^{\frac{1 + 5}{3}}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.3.4

Add $1$ and $5$ .

$f'' (x) = - \frac{\frac{2 (25 \cdot (3 {| 25 - x^{2} |}^{\frac{6}{3}}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.3.5

Divide $6$ by $3$ .

$f'' (x) = - \frac{\frac{2 (25 \cdot (3 {| 25 - x^{2} |}^{2}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.4

Multiply $25$ by $3$ .

$f'' (x) = - \frac{\frac{2 (75 {| 25 - x^{2} |}^{2} + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.5

Remove the absolute value in ${| 25 - x^{2} |}^{2}$ because exponentiations with even powers are always positive.

$f'' (x) = - \frac{\frac{2 (75 {(25 - x^{2})}^{2} + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.6

Rewrite ${(25 - x^{2})}^{2}$ as $(25 - x^{2}) (25 - x^{2})$ .

$f'' (x) = - \frac{\frac{2 (75 ((25 - x^{2}) (25 - x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.7

Expand $(25 - x^{2}) (25 - x^{2})$ using the FOIL Method.

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Step 2.30.5.9.7.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (75 (25 (25 - x^{2}) - x^{2} (25 - x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.7.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (75 (25 \cdot 25 + 25 (- x^{2}) - x^{2} (25 - x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.7.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (75 (25 \cdot 25 + 25 (- x^{2}) - x^{2} \cdot 25 - x^{2} (- x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8

Simplify and combine like terms.

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Step 2.30.5.9.8.1

Simplify each term.

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Step 2.30.5.9.8.1.1

Multiply $25$ by $25$ .

$f'' (x) = - \frac{\frac{2 (75 (625 + 25 (- x^{2}) - x^{2} \cdot 25 - x^{2} (- x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.2

Multiply $- 1$ by $25$ .

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - x^{2} \cdot 25 - x^{2} (- x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.3

Multiply $25$ by $- 1$ .

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - 25 x^{2} - x^{2} (- x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 x^{2} x^{2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.9.8.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 (x^{2} x^{2}))) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 x^{2 + 2})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - 25 x^{2} - 1 \cdot (- 1 x^{4})) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - 25 x^{2} + 1 x^{4}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.1.7

Multiply $x^{4}$ by $1$ .

$f'' (x) = - \frac{\frac{2 (75 (625 - 25 x^{2} - 25 x^{2} + x^{4}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.8.2

Subtract $25 x^{2}$ from $- 25 x^{2}$ .

$f'' (x) = - \frac{\frac{2 (75 (625 - 50 x^{2} + x^{4}) + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.9

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (75 \cdot 625 + 75 (- 50 x^{2}) + 75 x^{4} + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.10

Simplify.

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Step 2.30.5.9.10.1

Multiply $75$ by $625$ .

$f'' (x) = - \frac{\frac{2 (46875 + 75 (- 50 x^{2}) + 75 x^{4} + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.10.2

Multiply $- 50$ by $75$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} {(x + 5)}^{2} {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.11

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} ((x + 5) (x + 5)) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.12

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 2.30.5.9.12.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} (x (x + 5) + 5 (x + 5)) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.12.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} (x \cdot x + x \cdot 5 + 5 (x + 5)) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.12.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.13

Simplify and combine like terms.

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Step 2.30.5.9.13.1

Simplify each term.

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Step 2.30.5.9.13.1.1

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.13.1.2

Move $5$ to the left of $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.13.1.3

Multiply $5$ by $5$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} (x^{2} + 5 x + 5 x + 25) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.13.2

Add $5 x$ and $5 x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{2} (x^{2} + 10 x + 25) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.14

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{2} x^{2} + x^{2} (10 x) + x^{2} \cdot 25) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.15

Simplify.

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Step 2.30.5.9.15.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.9.15.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{2 + 2} + x^{2} (10 x) + x^{2} \cdot 25) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.15.1.2

Add $2$ and $2$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + x^{2} (10 x) + x^{2} \cdot 25) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.15.2

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{2} x + x^{2} \cdot 25) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.15.3

Move $25$ to the left of $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{2} x + 25 \cdot x^{2}) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.16

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.30.5.9.16.1

Move $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 (x \cdot x^{2}) + 25 \cdot x^{2}) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.16.2

Multiply $x$ by $x^{2}$ .

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Step 2.30.5.9.16.2.1

Raise $x$ to the power of $1$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 (x \cdot x^{2}) + 25 \cdot x^{2}) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.16.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{1 + 2} + 25 \cdot x^{2}) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.16.3

Add $1$ and $2$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 \cdot x^{2}) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) {(x - 5)}^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.17

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) ((x - 5) (x - 5)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.18

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 2.30.5.9.18.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) (x (x - 5) - 5 (x - 5)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.18.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) (x \cdot x + x \cdot - 5 - 5 (x - 5)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.18.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.19

Simplify and combine like terms.

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Step 2.30.5.9.19.1

Simplify each term.

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Step 2.30.5.9.19.1.1

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.19.1.2

Move $- 5$ to the left of $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.19.1.3

Multiply $- 5$ by $- 5$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) (x^{2} - 5 x - 5 x + 25))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.19.2

Subtract $5 x$ from $- 5 x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + (x^{4} + 10 x^{3} + 25 x^{2}) (x^{2} - 10 x + 25))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.20

Expand $(x^{4} + 10 x^{3} + 25 x^{2}) (x^{2} - 10 x + 25)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{4} x^{2} + x^{4} (- 10 x) + x^{4} \cdot 25 + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21

Simplify each term.

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Step 2.30.5.9.21.1

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.9.21.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{4 + 2} + x^{4} (- 10 x) + x^{4} \cdot 25 + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.1.2

Add $4$ and $2$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} + x^{4} (- 10 x) + x^{4} \cdot 25 + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.2

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{4} x + x^{4} \cdot 25 + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.3

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.30.5.9.21.3.1

Move $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 (x \cdot x^{4}) + x^{4} \cdot 25 + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.3.2

Multiply $x$ by $x^{4}$ .

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Step 2.30.5.9.21.3.2.1

Raise $x$ to the power of $1$ .

Step 2.30.5.9.21.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{1 + 4} + x^{4} \cdot 25 + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.3.3

Add $1$ and $4$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + x^{4} \cdot 25 + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.4

Move $25$ to the left of $x^{4}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 \cdot x^{4} + 10 x^{3} x^{2} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.9.21.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 (x^{2} x^{3}) + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{2 + 3} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.5.3

Add $2$ and $3$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} + 10 x^{3} (- 10 x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.6

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} + 10 \cdot (- 10 x^{3} x) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.7

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.30.5.9.21.7.1

Move $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} + 10 \cdot (- 10 (x \cdot x^{3})) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.7.2

Multiply $x$ by $x^{3}$ .

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Step 2.30.5.9.21.7.2.1

Raise $x$ to the power of $1$ .

Step 2.30.5.9.21.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} + 10 \cdot (- 10 x^{1 + 3}) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.7.3

Add $1$ and $3$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} + 10 \cdot (- 10 x^{4}) + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.8

Multiply $10$ by $- 10$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 10 x^{3} \cdot 25 + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.9

Multiply $25$ by $10$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{2} x^{2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.10

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.30.5.9.21.10.1

Move $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 (x^{2} x^{2}) + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.10.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{2 + 2} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.10.3

Add $2$ and $2$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} + 25 x^{2} (- 10 x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.11

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 x^{2} x) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.12

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.30.5.9.21.12.1

Move $x$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 (x \cdot x^{2})) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.12.2

Multiply $x$ by $x^{2}$ .

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Step 2.30.5.9.21.12.2.1

Raise $x$ to the power of $1$ .

Step 2.30.5.9.21.12.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 x^{1 + 2}) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.12.3

Add $1$ and $2$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} + 25 \cdot (- 10 x^{3}) + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.13

Multiply $25$ by $- 10$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} - 250 x^{3} + 25 x^{2} \cdot 25)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.21.14

Multiply $25$ by $25$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} - 250 x^{3} + 625 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.22

Combine the opposite terms in $x^{6} - 10 x^{5} + 25 x^{4} + 10 x^{5} - 100 x^{4} + 250 x^{3} + 25 x^{4} - 250 x^{3} + 625 x^{2}$ .

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Step 2.30.5.9.22.1

Add $- 10 x^{5}$ and $10 x^{5}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} + 25 x^{4} + 0 - 100 x^{4} + 250 x^{3} + 25 x^{4} - 250 x^{3} + 625 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.22.2

Add $x^{6} + 25 x^{4}$ and $0$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} + 25 x^{4} - 100 x^{4} + 250 x^{3} + 25 x^{4} - 250 x^{3} + 625 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.22.3

Subtract $250 x^{3}$ from $250 x^{3}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} + 25 x^{4} - 100 x^{4} + 25 x^{4} + 0 + 625 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.22.4

Add $x^{6} + 25 x^{4} - 100 x^{4} + 25 x^{4}$ and $0$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} + 25 x^{4} - 100 x^{4} + 25 x^{4} + 625 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.23

Subtract $100 x^{4}$ from $25 x^{4}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 75 x^{4} + 25 x^{4} + 625 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.24

Add $- 75 x^{4}$ and $25 x^{4}$ .

$f'' (x) = - \frac{\frac{2 (46875 - 3750 x^{2} + 75 x^{4} + x^{6} - 50 x^{4} + 625 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.25

Add $- 3750 x^{2}$ and $625 x^{2}$ .

$f'' (x) = - \frac{\frac{2 (46875 + 75 x^{4} + x^{6} - 50 x^{4} - 3125 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.26

Subtract $50 x^{4}$ from $75 x^{4}$ .

$f'' (x) = - \frac{\frac{2 (46875 + x^{6} + 25 x^{4} - 3125 x^{2})}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.27

Reorder terms.

$f'' (x) = - \frac{\frac{2 (x^{6} + 25 x^{4} - 3125 x^{2} + 46875)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28

Rewrite $x^{6} + 25 x^{4} - 3125 x^{2} + 46875$ in a factored form.

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Step 2.30.5.9.28.1

Factor $x^{6} + 25 x^{4} - 3125 x^{2} + 46875$ using the rational roots test.

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Step 2.30.5.9.28.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 46875, \pm 3, \pm 15625, \pm 5, \pm 9375, \pm 15, \pm 3125, \pm 25, \pm 1875, \pm 75, \pm 625, \pm 125, \pm 375$

$q = \pm 1$

Step 2.30.5.9.28.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 46875, \pm 3, \pm 15625, \pm 5, \pm 9375, \pm 15, \pm 3125, \pm 25, \pm 1875, \pm 75, \pm 625, \pm 125, \pm 375$

Step 2.30.5.9.28.1.3

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

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Step 2.30.5.9.28.1.3.1

Substitute $- 5$ into the polynomial.

${(- 5)}^{6} + 25 {(- 5)}^{4} - 3125 {(- 5)}^{2} + 46875$

Step 2.30.5.9.28.1.3.2

Raise $- 5$ to the power of $6$ .

$15625 + 25 {(- 5)}^{4} - 3125 {(- 5)}^{2} + 46875$

Step 2.30.5.9.28.1.3.3

Raise $- 5$ to the power of $4$ .

$15625 + 25 \cdot 625 - 3125 {(- 5)}^{2} + 46875$

Step 2.30.5.9.28.1.3.4

Multiply $25$ by $625$ .

$15625 + 15625 - 3125 {(- 5)}^{2} + 46875$

Step 2.30.5.9.28.1.3.5

Add $15625$ and $15625$ .

$31250 - 3125 {(- 5)}^{2} + 46875$

Step 2.30.5.9.28.1.3.6

Raise $- 5$ to the power of $2$ .

$31250 - 3125 \cdot 25 + 46875$

Step 2.30.5.9.28.1.3.7

Multiply $- 3125$ by $25$ .

$31250 - 78125 + 46875$

Step 2.30.5.9.28.1.3.8

Subtract $78125$ from $31250$ .

$- 46875 + 46875$

Step 2.30.5.9.28.1.3.9

Add $- 46875$ and $46875$ .

$0$

Step 2.30.5.9.28.1.4

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{6} + 25 x^{4} - 3125 x^{2} + 46875}{x + 5}$

Step 2.30.5.9.28.1.5

Divide $x^{6} + 25 x^{4} - 3125 x^{2} + 46875$ by $x + 5$ .

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Step 2.30.5.9.28.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

Step 2.30.5.9.28.1.5.2

Divide the highest order term in the dividend $x^{6}$ by the highest order term in divisor $x$ .

$x^{5}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

Step 2.30.5.9.28.1.5.3

Multiply the new quotient term by the divisor.

$x^{5}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

Step 2.30.5.9.28.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{6} + 5 x^{5}$

$x^{5}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

Step 2.30.5.9.28.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

Step 2.30.5.9.28.1.5.6

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

Step 2.30.5.9.28.1.5.7

Divide the highest order term in the dividend $- 5 x^{5}$ by the highest order term in divisor $x$ .

$x^{5}$

$5 x^{4}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

Step 2.30.5.9.28.1.5.8

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

Step 2.30.5.9.28.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{5} - 25 x^{4}$

$x^{5}$

$5 x^{4}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

Step 2.30.5.9.28.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

Step 2.30.5.9.28.1.5.11

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

Step 2.30.5.9.28.1.5.12

Divide the highest order term in the dividend $50 x^{4}$ by the highest order term in divisor $x$ .

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

Step 2.30.5.9.28.1.5.13

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

Step 2.30.5.9.28.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $50 x^{4} + 250 x^{3}$

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

Step 2.30.5.9.28.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

Step 2.30.5.9.28.1.5.16

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

Step 2.30.5.9.28.1.5.17

Divide the highest order term in the dividend $- 250 x^{3}$ by the highest order term in divisor $x$ .

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

Step 2.30.5.9.28.1.5.18

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

Step 2.30.5.9.28.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 250 x^{3} - 1250 x^{2}$

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

Step 2.30.5.9.28.1.5.20

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

Step 2.30.5.9.28.1.5.21

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

Step 2.30.5.9.28.1.5.22

Divide the highest order term in the dividend $- 1875 x^{2}$ by the highest order term in divisor $x$ .

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

Step 2.30.5.9.28.1.5.23

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

Step 2.30.5.9.28.1.5.24

The expression needs to be subtracted from the dividend, so change all the signs in $- 1875 x^{2} - 9375 x$

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

Step 2.30.5.9.28.1.5.25

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

Step 2.30.5.9.28.1.5.26

Pull the next terms from the original dividend down into the current dividend.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

$46875$

Step 2.30.5.9.28.1.5.27

Divide the highest order term in the dividend $9375 x$ by the highest order term in divisor $x$ .

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$9375$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

$46875$

Step 2.30.5.9.28.1.5.28

Multiply the new quotient term by the divisor.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$9375$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

$46875$

$9375 x$

$46875$

Step 2.30.5.9.28.1.5.29

The expression needs to be subtracted from the dividend, so change all the signs in $9375 x + 46875$

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$9375$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

$46875$

$9375 x$

$46875$

Step 2.30.5.9.28.1.5.30

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{5}$

$5 x^{4}$

$50 x^{3}$

$250 x^{2}$

$1875 x$

$9375$

$x$

$5$

$x^{6}$

$0 x^{5}$

$25 x^{4}$

$0 x^{3}$

$3125 x^{2}$

$0 x$

$46875$

$x^{6}$

$5 x^{5}$

$25 x^{4}$

$5 x^{5}$

$25 x^{4}$

$50 x^{4}$

$0 x^{3}$

$50 x^{4}$

$250 x^{3}$

$3125 x^{2}$

$250 x^{3}$

$1250 x^{2}$

$1875 x^{2}$

$0 x$

$1875 x^{2}$

$9375 x$

$46875$

$9375 x$

$46875$

$0$

Step 2.30.5.9.28.1.5.31

Since the remander is $0$ , the final answer is the quotient.

$x^{5} - 5 x^{4} + 50 x^{3} - 250 x^{2} - 1875 x + 9375$

Step 2.30.5.9.28.1.6

Write $x^{6} + 25 x^{4} - 3125 x^{2} + 46875$ as a set of factors.

$f'' (x) = - \frac{\frac{2 ((x + 5) (x^{5} - 5 x^{4} + 50 x^{3} - 250 x^{2} - 1875 x + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.2

Regroup terms.

$f'' (x) = - \frac{\frac{2 ((x + 5) (x^{5} + 50 x^{3} - 1875 x - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.3

Factor $x$ out of $x^{5} + 50 x^{3} - 1875 x$ .

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Step 2.30.5.9.28.3.1

Factor $x$ out of $x^{5}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x \cdot x^{4} + 50 x^{3} - 1875 x - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.3.2

Factor $x$ out of $50 x^{3}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x \cdot x^{4} + x (50 x^{2}) - 1875 x - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.3.3

Factor $x$ out of $- 1875 x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x \cdot x^{4} + x (50 x^{2}) + x \cdot - 1875 - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.3.4

Factor $x$ out of $x \cdot x^{4} + x (50 x^{2})$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x^{4} + 50 x^{2}) + x \cdot - 1875 - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.3.5

Factor $x$ out of $x (x^{4} + 50 x^{2}) + x \cdot - 1875$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x^{4} + 50 x^{2} - 1875) - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.4

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x ({(x^{2})}^{2} + 50 x^{2} - 1875) - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.5

Let $u_{3} = x^{2}$ . Substitute $u_{3}$ for all occurrences of $x^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x ({u_{3}}^{2} + 50 u_{3} - 1875) - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.6

Factor ${u_{3}}^{2} + 50 u_{3} - 1875$ using the AC method.

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Step 2.30.5.9.28.6.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 1875$ and whose sum is $50$ .

$- 25, 75$

Step 2.30.5.9.28.6.2

Write the factored form using these integers.

$f'' (x) = - \frac{\frac{2 ((x + 5) (x ((u_{3} - 25) (u_{3} + 75)) - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.7

Replace all occurrences of $u_{3}$ with $x^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x ((x^{2} - 25) (x^{2} + 75)) - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.8

Rewrite $25$ as $5^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x ((x^{2} - 5^{2}) (x^{2} + 75)) - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.9

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) - 5 x^{4} - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.10

Factor $5$ out of $- 5 x^{4} - 250 x^{2} + 9375$ .

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Step 2.30.5.9.28.10.1

Factor $5$ out of $- 5 x^{4}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- x^{4}) - 250 x^{2} + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.10.2

Factor $5$ out of $- 250 x^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- x^{4}) + 5 (- 50 x^{2}) + 9375))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.10.3

Factor $5$ out of $9375$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- x^{4}) + 5 (- 50 x^{2}) + 5 (1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.10.4

Factor $5$ out of $5 (- x^{4}) + 5 (- 50 x^{2})$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- x^{4} - 50 x^{2}) + 5 (1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.10.5

Factor $5$ out of $5 (- x^{4} - 50 x^{2}) + 5 (1875)$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- x^{4} - 50 x^{2} + 1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.11

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- {(x^{2})}^{2} - 50 x^{2} + 1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.12

Let $u_{4} = x^{2}$ . Substitute $u_{4}$ for all occurrences of $x^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- {u_{4}}^{2} - 50 u_{4} + 1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.13

Factor by grouping.

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Step 2.30.5.9.28.13.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 1875 = - 1875$ and whose sum is $b = - 50$ .

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Step 2.30.5.9.28.13.1.1

Factor $- 50$ out of $- 50 u_{4}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- {u_{4}}^{2} - 50 u_{4} + 1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.13.1.2

Rewrite $- 50$ as $25$ plus $- 75$

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- {u_{4}}^{2} + (25 - 75) u_{4} + 1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.13.1.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (- {u_{4}}^{2} + 25 u_{4} - 75 u_{4} + 1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.13.2

Factor out the greatest common factor from each group.

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Step 2.30.5.9.28.13.2.1

Group the first two terms and the last two terms.

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 ((- {u_{4}}^{2} + 25 u_{4}) - 75 u_{4} + 1875)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.13.2.2

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (u_{4} (- u_{4} + 25) + 75 (- u_{4} + 25))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.13.3

Factor the polynomial by factoring out the greatest common factor, $- u_{4} + 25$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 ((- u_{4} + 25) (u_{4} + 75))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.14

Replace all occurrences of $u_{4}$ with $x^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 ((- x^{2} + 25) (x^{2} + 75))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.15

Rewrite $25$ as $5^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 ((- x^{2} + 5^{2}) (x^{2} + 75))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.16

Reorder $- x^{2}$ and $5^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 ((5^{2} - x^{2}) (x^{2} + 75))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.17

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x (x + 5) (x - 5) (x^{2} + 75) + 5 (5 + x) (5 - x) (x^{2} + 75)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.18

Factor $x^{2} + 75$ out of $x (x + 5) (x - 5) (x^{2} + 75) + 5 (5 + x) (5 - x) (x^{2} + 75)$ .

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Step 2.30.5.9.28.18.1

Factor $x^{2} + 75$ out of $x (x + 5) (x - 5) (x^{2} + 75)$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x (x + 5) (x - 5)) + 5 (5 + x) (5 - x) (x^{2} + 75)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.18.2

Factor $x^{2} + 75$ out of $5 (5 + x) (5 - x) (x^{2} + 75)$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x (x + 5) (x - 5)) + (x^{2} + 75) (5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.18.3

Factor $x^{2} + 75$ out of $(x^{2} + 75) (x (x + 5) (x - 5)) + (x^{2} + 75) (5 (5 + x) (5 - x))$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x (x + 5) (x - 5) + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.19

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) ((x \cdot x + x \cdot 5) (x - 5) + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.20

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) ((x^{2} + x \cdot 5) (x - 5) + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.21

Move $5$ to the left of $x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) ((x^{2} + 5 \cdot x) (x - 5) + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.22

Expand $(x^{2} + 5 x) (x - 5)$ using the FOIL Method.

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Step 2.30.5.9.28.22.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{2} (x - 5) + 5 x (x - 5) + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.22.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{2} x + x^{2} \cdot - 5 + 5 x (x - 5) + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.22.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{2} x + x^{2} \cdot - 5 + 5 x \cdot x + 5 x \cdot - 5 + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23

Simplify and combine like terms.

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Step 2.30.5.9.28.23.1

Simplify each term.

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Step 2.30.5.9.28.23.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.30.5.9.28.23.1.1.1

Multiply $x^{2}$ by $x$ .

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Step 2.30.5.9.28.23.1.1.1.1

Raise $x$ to the power of $1$ .

Step 2.30.5.9.28.23.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{2 + 1} + x^{2} \cdot - 5 + 5 x \cdot x + 5 x \cdot - 5 + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23.1.1.2

Add $2$ and $1$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} + x^{2} \cdot - 5 + 5 x \cdot x + 5 x \cdot - 5 + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23.1.2

Move $- 5$ to the left of $x^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 5 \cdot x^{2} + 5 x \cdot x + 5 x \cdot - 5 + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.30.5.9.28.23.1.3.1

Move $x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 5 x^{2} + 5 (x \cdot x) + 5 x \cdot - 5 + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23.1.3.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 5 x^{2} + 5 x^{2} + 5 x \cdot - 5 + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23.1.4

Multiply $- 5$ by $5$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 5 x^{2} + 5 x^{2} - 25 x + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23.2

Add $- 5 x^{2}$ and $5 x^{2}$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} + 0 - 25 x + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.23.3

Add $x^{3}$ and $0$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 5 (5 + x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.24

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + (5 \cdot 5 + 5 x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.25

Multiply $5$ by $5$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + (25 + 5 x) (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.26

Expand $(25 + 5 x) (5 - x)$ using the FOIL Method.

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Step 2.30.5.9.28.26.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 25 (5 - x) + 5 x (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.26.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 25 \cdot 5 + 25 (- x) + 5 x (5 - x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.26.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 25 \cdot 5 + 25 (- x) + 5 x \cdot 5 + 5 x (- x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27

Simplify and combine like terms.

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Step 2.30.5.9.28.27.1

Simplify each term.

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Step 2.30.5.9.28.27.1.1

Multiply $25$ by $5$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 + 25 (- x) + 5 x \cdot 5 + 5 x (- x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.1.2

Multiply $- 1$ by $25$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 - 25 x + 5 x \cdot 5 + 5 x (- x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.1.3

Multiply $5$ by $5$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 - 25 x + 25 x + 5 x (- x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 - 25 x + 25 x + 5 \cdot (- 1 x \cdot x))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.30.5.9.28.27.1.5.1

Move $x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 - 25 x + 25 x + 5 \cdot (- 1 (x \cdot x)))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.1.5.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 - 25 x + 25 x + 5 \cdot (- 1 x^{2}))))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.1.6

Multiply $5$ by $- 1$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 - 25 x + 25 x - 5 x^{2})))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.2

Add $- 25 x$ and $25 x$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 + 0 - 5 x^{2})))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.27.3

Add $125$ and $0$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 25 x + 125 - 5 x^{2})))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.28

Reorder terms.

$f'' (x) = - \frac{\frac{2 ((x + 5) ((x^{2} + 75) (x^{3} - 5 x^{2} - 25 x + 125)))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.28.29

Factor.

$f'' (x) = - \frac{\frac{2 ((x + 5) (x^{2} + 75) {(x - 5)}^{2} (x + 5))}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

$f'' (x) = - \frac{\frac{2 (x + 5) (x^{2} + 75) {(x - 5)}^{2} (x + 5)}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.29

Combine exponents.

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Step 2.30.5.9.29.1

Raise $x + 5$ to the power of $1$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x + 5)) (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.29.2

Raise $x + 5$ to the power of $1$ .

$f'' (x) = - \frac{\frac{2 ((x + 5) (x + 5)) (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.29.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{\frac{2 {(x + 5)}^{1 + 1} (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.5.9.29.4

Add $1$ and $1$ .

$f'' (x) = - \frac{\frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.6

Combine terms.

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Step 2.30.6.1

Rewrite $\frac{\frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$ as a product.

$f'' (x) = - (\frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}} \cdot \frac{1}{3 {| 25 - x^{2} |}^{\frac{10}{3}}})$

Step 2.30.6.2

Multiply $\frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}}}$ by $\frac{1}{3 {| 25 - x^{2} |}^{\frac{10}{3}}}$ .

$f'' (x) = - \frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{3 {| 25 - x^{2} |}^{\frac{1}{3}} (3 {| 25 - x^{2} |}^{\frac{10}{3}})}$

Step 2.30.6.3

Multiply $3$ by $3$ .

$f'' (x) = - \frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{9 {| 25 - x^{2} |}^{\frac{1}{3}} {| 25 - x^{2} |}^{\frac{10}{3}}}$

Step 2.30.6.4

Multiply ${| 25 - x^{2} |}^{\frac{1}{3}}$ by ${| 25 - x^{2} |}^{\frac{10}{3}}$ by adding the exponents.

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Step 2.30.6.4.1

Move ${| 25 - x^{2} |}^{\frac{10}{3}}$ .

$f'' (x) = - \frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{9 ({| 25 - x^{2} |}^{\frac{10}{3}} {| 25 - x^{2} |}^{\frac{1}{3}})}$

Step 2.30.6.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{9 {| 25 - x^{2} |}^{\frac{10}{3} + \frac{1}{3}}}$

Step 2.30.6.4.3

Combine the numerators over the common denominator.

$f'' (x) = - \frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{9 {| 25 - x^{2} |}^{\frac{10 + 1}{3}}}$

Step 2.30.6.4.4

Add $10$ and $1$ .

$f'' (x) = - \frac{2 {(x + 5)}^{2} (x^{2} + 75) {(x - 5)}^{2}}{9 {| 25 - x^{2} |}^{\frac{11}{3}}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $\sqrt[3]{| 25 - x^{2} |} + 6$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{| 25 - x^{2} |}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [\sqrt[3]{| 25 - x^{2} |}] + \frac{d}{d x} [6]$

Step 4.1.2

Evaluate $\frac{d}{d x} [\sqrt[3]{| 25 - x^{2} |}]$ .

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Step 4.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{| 25 - x^{2} |}$ as ${| 25 - x^{2} |}^{\frac{1}{3}}$ .

$\frac{d}{d x} [{| 25 - x^{2} |}^{\frac{1}{3}}] + \frac{d}{d x} [6]$

Step 4.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = | 25 - x^{2} |$ .

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Step 4.1.2.2.1

To apply the Chain Rule, set $u_{1}$ as $| 25 - x^{2} |$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{3}}] \frac{d}{d x} [| 25 - x^{2} |] + \frac{d}{d x} [6]$

Step 4.1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{3}$ .

$\frac{1}{3} {u_{1}}^{\frac{1}{3} - 1} \frac{d}{d x} [| 25 - x^{2} |] + \frac{d}{d x} [6]$

Step 4.1.2.2.3

Replace all occurrences of $u_{1}$ with $| 25 - x^{2} |$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} \frac{d}{d x} [| 25 - x^{2} |] + \frac{d}{d x} [6]$

Step 4.1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = 25 - x^{2}$ .

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Step 4.1.2.3.1

To apply the Chain Rule, set $u_{2}$ as $25 - x^{2}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{d}{d u_{2}} [| u_{2} |] \frac{d}{d x} [25 - x^{2}]) + \frac{d}{d x} [6]$

Step 4.1.2.3.2

The derivative of $| u_{2} |$ with respect to $u_{2}$ is $\frac{u_{2}}{| u_{2} |}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{u_{2}}{| u_{2} |} \frac{d}{d x} [25 - x^{2}]) + \frac{d}{d x} [6]$

Step 4.1.2.3.3

Replace all occurrences of $u_{2}$ with $25 - x^{2}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} \frac{d}{d x} [25 - x^{2}]) + \frac{d}{d x} [6]$

Step 4.1.2.4

By the Sum Rule, the derivative of $25 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (\frac{d}{d x} [25] + \frac{d}{d x} [- x^{2}])) + \frac{d}{d x} [6]$

Step 4.1.2.5

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 + \frac{d}{d x} [- x^{2}])) + \frac{d}{d x} [6]$

Step 4.1.2.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - \frac{d}{d x} [x^{2}])) + \frac{d}{d x} [6]$

Step 4.1.2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 4.1.2.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 4.1.2.9

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 4.1.2.10

Combine the numerators over the common denominator.

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1 - 1 \cdot 3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 4.1.2.11

Simplify the numerator.

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Step 4.1.2.11.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{1 - 3}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 4.1.2.11.2

Subtract $3$ from $1$ .

$\frac{1}{3} {| 25 - x^{2} |}^{\frac{- 2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 4.1.2.12

Move the negative in front of the fraction.

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - (2 x))) + \frac{d}{d x} [6]$

Step 4.1.2.13

Multiply $2$ by $- 1$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (0 - 2 x)) + \frac{d}{d x} [6]$

Step 4.1.2.14

Subtract $2 x$ from $0$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{25 - x^{2}}{| 25 - x^{2} |} (- 2 x)) + \frac{d}{d x} [6]$

Step 4.1.2.15

Combine $- 2$ and $\frac{25 - x^{2}}{| 25 - x^{2} |}$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (\frac{- 2 (25 - x^{2})}{| 25 - x^{2} |} x) + \frac{d}{d x} [6]$

Step 4.1.2.16

Combine $\frac{- 2 (25 - x^{2})}{| 25 - x^{2} |}$ and $x$ .

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} \frac{- 2 (25 - x^{2}) x}{| 25 - x^{2} |} + \frac{d}{d x} [6]$

Step 4.1.2.17

Move the negative in front of the fraction.

$\frac{1}{3} {| 25 - x^{2} |}^{- \frac{2}{3}} (- \frac{(2 (25 - x^{2})) x}{| 25 - x^{2} |}) + \frac{d}{d x} [6]$

Step 4.1.2.18

Combine $\frac{1}{3}$ and ${| 25 - x^{2} |}^{- \frac{2}{3}}$ .

$\frac{{| 25 - x^{2} |}^{- \frac{2}{3}}}{3} (- \frac{(2 (25 - x^{2})) x}{| 25 - x^{2} |}) + \frac{d}{d x} [6]$

Step 4.1.2.19

Multiply $\frac{{| 25 - x^{2} |}^{- \frac{2}{3}}}{3}$ by $\frac{(2 (25 - x^{2})) x}{| 25 - x^{2} |}$ .

$- \frac{{| 25 - x^{2} |}^{- \frac{2}{3}} ((2 (25 - x^{2})) x)}{3 | 25 - x^{2} |} + \frac{d}{d x} [6]$

Step 4.1.2.20

Move ${| 25 - x^{2} |}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{(2 (25 - x^{2})) x}{3 | 25 - x^{2} | {| 25 - x^{2} |}^{\frac{2}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.21

Multiply $| 25 - x^{2} |$ by ${| 25 - x^{2} |}^{\frac{2}{3}}$ by adding the exponents.

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Step 4.1.2.21.1

Move ${| 25 - x^{2} |}^{\frac{2}{3}}$ .

$- \frac{(2 (25 - x^{2})) x}{3 ({| 25 - x^{2} |}^{\frac{2}{3}} | 25 - x^{2} |)} + \frac{d}{d x} [6]$

Step 4.1.2.21.2

Multiply ${| 25 - x^{2} |}^{\frac{2}{3}}$ by $| 25 - x^{2} |$ .

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Step 4.1.2.21.2.1

Raise $| 25 - x^{2} |$ to the power of $1$ .

$- \frac{(2 (25 - x^{2})) x}{3 ({| 25 - x^{2} |}^{\frac{2}{3}} {| 25 - x^{2} |}^{1})} + \frac{d}{d x} [6]$

Step 4.1.2.21.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{2}{3} + 1}} + \frac{d}{d x} [6]$

Step 4.1.2.21.3

Write $1$ as a fraction with a common denominator.

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{2}{3} + \frac{3}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.21.4

Combine the numerators over the common denominator.

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{2 + 3}{3}}} + \frac{d}{d x} [6]$

Step 4.1.2.21.5

Add $2$ and $3$ .

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + \frac{d}{d x} [6]$

Step 4.1.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$- \frac{(2 (25 - x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Apply the distributive property.

$- \frac{(2 \cdot 25 + 2 (- x^{2})) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4.2

Apply the distributive property.

$- \frac{2 \cdot 25 x + 2 (- x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4.3

Combine terms.

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Step 4.1.4.3.1

Multiply $2$ by $25$ .

$- \frac{50 x + 2 (- x^{2}) x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4.3.2

Multiply $- 1$ by $2$ .

$- \frac{50 x - 2 x^{2} x}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4.3.3

Raise $x$ to the power of $1$ .

$- \frac{50 x - 2 (x^{1} x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{50 x - 2 x^{1 + 2}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4.3.5

Add $1$ and $2$ .

$- \frac{50 x - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} + 0$

Step 4.1.4.3.6

Add $- \frac{50 x - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ and $0$ .

$- \frac{50 x - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 4.1.4.4

Simplify the numerator.

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Step 4.1.4.4.1

Factor $2 x$ out of $50 x - 2 x^{3}$ .

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Step 4.1.4.4.1.1

Factor $2 x$ out of $50 x$ .

$- \frac{2 x (25) - 2 x^{3}}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 4.1.4.4.1.2

Factor $2 x$ out of $- 2 x^{3}$ .

$- \frac{2 x (25) + 2 x (- x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 4.1.4.4.1.3

Factor $2 x$ out of $2 x (25) + 2 x (- x^{2})$ .

$- \frac{2 x (25 - x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 4.1.4.4.2

Rewrite $25$ as $5^{2}$ .

$- \frac{2 x (5^{2} - x^{2})}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 4.1.4.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$f' (x) = - \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ .

$- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} = 0$

Step 5.2

Set the numerator equal to zero.

$2 x (5 + x) (5 - x) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$5 + x = 0$

$5 - x = 0$

Step 5.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.3.3

Set $5 + x$ equal to $0$ and solve for $x$ .

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Step 5.3.3.1

Set $5 + x$ equal to $0$ .

$5 + x = 0$

Step 5.3.3.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 5.3.4

Set $5 - x$ equal to $0$ and solve for $x$ .

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Step 5.3.4.1

Set $5 - x$ equal to $0$ .

$5 - x = 0$

Step 5.3.4.2

Solve $5 - x = 0$ for $x$ .

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Step 5.3.4.2.1

Subtract $5$ from both sides of the equation.

$- x = - 5$

Step 5.3.4.2.2

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 5.3.4.2.2.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 5.3.4.2.2.2

Simplify the left side.

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Step 5.3.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 5.3.4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 5.3.4.2.2.3

Simplify the right side.

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Step 5.3.4.2.2.3.1

Divide $- 5$ by $- 1$ .

$x = 5$

Step 5.3.5

The final solution is all the values that make $2 x (5 + x) (5 - x) = 0$ true.

$x = 0, - 5, 5$

Step 5.4

Exclude the solutions that do not make $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}} = 0$ true.

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{2 x (5 + x) (5 - x)}{3 \sqrt[3]{{| 25 - x^{2} |}^{5}}}$

Step 6.2

Set the denominator in $\frac{2 x (5 + x) (5 - x)}{3 \sqrt[3]{{| 25 - x^{2} |}^{5}}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{{| 25 - x^{2} |}^{5}} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Simplify both sides of the equation.

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Step 6.3.1.1

Rewrite $25$ as $5^{2}$ .

$3 \sqrt[3]{{| 5^{2} - x^{2} |}^{5}} = 0$

Step 6.3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$3 \sqrt[3]{{| (5 + x) (5 - x) |}^{5}} = 0$

Step 6.3.1.3

Expand $(5 + x) (5 - x)$ using the FOIL Method.

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Step 6.3.1.3.1

Apply the distributive property.

$3 \sqrt[3]{{| 5 (5 - x) + x (5 - x) |}^{5}} = 0$

Step 6.3.1.3.2

Apply the distributive property.

$3 \sqrt[3]{{| 5 \cdot 5 + 5 (- x) + x (5 - x) |}^{5}} = 0$

Step 6.3.1.3.3

Apply the distributive property.

$3 \sqrt[3]{{| 5 \cdot 5 + 5 (- x) + x \cdot 5 + x (- x) |}^{5}} = 0$

Step 6.3.1.4

Simplify and combine like terms.

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Step 6.3.1.4.1

Simplify each term.

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Step 6.3.1.4.1.1

Multiply $5$ by $5$ .

$3 \sqrt[3]{{| 25 + 5 (- x) + x \cdot 5 + x (- x) |}^{5}} = 0$

Step 6.3.1.4.1.2

Multiply $- 1$ by $5$ .

$3 \sqrt[3]{{| 25 - 5 x + x \cdot 5 + x (- x) |}^{5}} = 0$

Step 6.3.1.4.1.3

Move $5$ to the left of $x$ .

$3 \sqrt[3]{{| 25 - 5 x + 5 \cdot x + x (- x) |}^{5}} = 0$

Step 6.3.1.4.1.4

Rewrite using the commutative property of multiplication.

$3 \sqrt[3]{{| 25 - 5 x + 5 x - x \cdot x |}^{5}} = 0$

Step 6.3.1.4.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.4.1.5.1

Move $x$ .

$3 \sqrt[3]{{| 25 - 5 x + 5 x - (x \cdot x) |}^{5}} = 0$

Step 6.3.1.4.1.5.2

Multiply $x$ by $x$ .

$3 \sqrt[3]{{| 25 - 5 x + 5 x - x^{2} |}^{5}} = 0$

Step 6.3.1.4.2

Add $- 5 x$ and $5 x$ .

$3 \sqrt[3]{{| 25 + 0 - x^{2} |}^{5}} = 0$

Step 6.3.1.4.3

Add $25$ and $0$ .

$3 \sqrt[3]{{| 25 - x^{2} |}^{5}} = 0$

Step 6.3.1.5

Rewrite $25$ as $5^{2}$ .

$3 \sqrt[3]{{| 5^{2} - x^{2} |}^{5}} = 0$

Step 6.3.1.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$3 \sqrt[3]{{| (5 + x) (5 - x) |}^{5}} = 0$

Step 6.3.1.7

Factor out ${| (5 + x) (5 - x) |}^{3}$ .

$3 \sqrt[3]{{| (5 + x) (5 - x) |}^{3} {| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.8

Pull terms out from under the radical.

$3 (| (5 + x) (5 - x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}}) = 0$

Step 6.3.1.9

Expand $(5 + x) (5 - x)$ using the FOIL Method.

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Step 6.3.1.9.1

Apply the distributive property.

$3 | 5 (5 - x) + x (5 - x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.9.2

Apply the distributive property.

$3 | 5 \cdot 5 + 5 (- x) + x (5 - x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.9.3

Apply the distributive property.

$3 | 5 \cdot 5 + 5 (- x) + x \cdot 5 + x (- x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10

Simplify and combine like terms.

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Step 6.3.1.10.1

Simplify each term.

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Step 6.3.1.10.1.1

Multiply $5$ by $5$ .

$3 | 25 + 5 (- x) + x \cdot 5 + x (- x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10.1.2

Multiply $- 1$ by $5$ .

$3 | 25 - 5 x + x \cdot 5 + x (- x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10.1.3

Move $5$ to the left of $x$ .

$3 | 25 - 5 x + 5 \cdot x + x (- x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10.1.4

Rewrite using the commutative property of multiplication.

$3 | 25 - 5 x + 5 x - x \cdot x | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.10.1.5.1

Move $x$ .

$3 | 25 - 5 x + 5 x - (x \cdot x) | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10.1.5.2

Multiply $x$ by $x$ .

$3 | 25 - 5 x + 5 x - x^{2} | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10.2

Add $- 5 x$ and $5 x$ .

$3 | 25 + 0 - x^{2} | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.10.3

Add $25$ and $0$ .

$3 | 25 - x^{2} | \sqrt[3]{{| (5 + x) (5 - x) |}^{2}} = 0$

Step 6.3.1.11

Expand $(5 + x) (5 - x)$ using the FOIL Method.

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Step 6.3.1.11.1

Apply the distributive property.

$3 | 25 - x^{2} | \sqrt[3]{{| 5 (5 - x) + x (5 - x) |}^{2}} = 0$

Step 6.3.1.11.2

Apply the distributive property.

$3 | 25 - x^{2} | \sqrt[3]{{| 5 \cdot 5 + 5 (- x) + x (5 - x) |}^{2}} = 0$

Step 6.3.1.11.3

Apply the distributive property.

$3 | 25 - x^{2} | \sqrt[3]{{| 5 \cdot 5 + 5 (- x) + x \cdot 5 + x (- x) |}^{2}} = 0$

Step 6.3.1.12

Simplify and combine like terms.

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Step 6.3.1.12.1

Simplify each term.

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Step 6.3.1.12.1.1

Multiply $5$ by $5$ .

$3 | 25 - x^{2} | \sqrt[3]{{| 25 + 5 (- x) + x \cdot 5 + x (- x) |}^{2}} = 0$

Step 6.3.1.12.1.2

Multiply $- 1$ by $5$ .

$3 | 25 - x^{2} | \sqrt[3]{{| 25 - 5 x + x \cdot 5 + x (- x) |}^{2}} = 0$

Step 6.3.1.12.1.3

Move $5$ to the left of $x$ .

$3 | 25 - x^{2} | \sqrt[3]{{| 25 - 5 x + 5 \cdot x + x (- x) |}^{2}} = 0$

Step 6.3.1.12.1.4

Rewrite using the commutative property of multiplication.

$3 | 25 - x^{2} | \sqrt[3]{{| 25 - 5 x + 5 x - x \cdot x |}^{2}} = 0$

Step 6.3.1.12.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 6.3.1.12.1.5.1

Move $x$ .

$3 | 25 - x^{2} | \sqrt[3]{{| 25 - 5 x + 5 x - (x \cdot x) |}^{2}} = 0$

Step 6.3.1.12.1.5.2

Multiply $x$ by $x$ .

$3 | 25 - x^{2} | \sqrt[3]{{| 25 - 5 x + 5 x - x^{2} |}^{2}} = 0$

Step 6.3.1.12.2

Add $- 5 x$ and $5 x$ .

$3 | 25 - x^{2} | \sqrt[3]{{| 25 + 0 - x^{2} |}^{2}} = 0$

Step 6.3.1.12.3

Add $25$ and $0$ .

$3 | 25 - x^{2} | \sqrt[3]{{| 25 - x^{2} |}^{2}} = 0$

Step 6.3.1.13

Remove the absolute value in ${| 25 - x^{2} |}^{2}$ because exponentiations with even powers are always positive.

$3 | 25 - x^{2} | \sqrt[3]{{(25 - x^{2})}^{2}} = 0$

Step 6.3.1.14

Rewrite ${(25 - x^{2})}^{2}$ as $(25 - x^{2}) (25 - x^{2})$ .

$3 | 25 - x^{2} | \sqrt[3]{(25 - x^{2}) (25 - x^{2})} = 0$

Step 6.3.1.15

Expand $(25 - x^{2}) (25 - x^{2})$ using the FOIL Method.

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Step 6.3.1.15.1

Apply the distributive property.

$3 | 25 - x^{2} | \sqrt[3]{25 (25 - x^{2}) - x^{2} (25 - x^{2})} = 0$

Step 6.3.1.15.2

Apply the distributive property.

$3 | 25 - x^{2} | \sqrt[3]{25 \cdot 25 + 25 (- x^{2}) - x^{2} (25 - x^{2})} = 0$

Step 6.3.1.15.3

Apply the distributive property.

$3 | 25 - x^{2} | \sqrt[3]{25 \cdot 25 + 25 (- x^{2}) - x^{2} \cdot 25 - x^{2} (- x^{2})} = 0$

Step 6.3.1.16

Simplify and combine like terms.

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Step 6.3.1.16.1

Simplify each term.

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Step 6.3.1.16.1.1

Multiply $25$ by $25$ .

$3 | 25 - x^{2} | \sqrt[3]{625 + 25 (- x^{2}) - x^{2} \cdot 25 - x^{2} (- x^{2})} = 0$

Step 6.3.1.16.1.2

Multiply $- 1$ by $25$ .

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - x^{2} \cdot 25 - x^{2} (- x^{2})} = 0$

Step 6.3.1.16.1.3

Multiply $25$ by $- 1$ .

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - 25 x^{2} - x^{2} (- x^{2})} = 0$

Step 6.3.1.16.1.4

Rewrite using the commutative property of multiplication.

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - 25 x^{2} - 1 \cdot - 1 x^{2} x^{2}} = 0$

Step 6.3.1.16.1.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 6.3.1.16.1.5.1

Move $x^{2}$ .

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - 25 x^{2} - 1 \cdot - 1 (x^{2} x^{2})} = 0$

Step 6.3.1.16.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - 25 x^{2} - 1 \cdot - 1 x^{2 + 2}} = 0$

Step 6.3.1.16.1.5.3

Add $2$ and $2$ .

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - 25 x^{2} - 1 \cdot - 1 x^{4}} = 0$

Step 6.3.1.16.1.6

Multiply $- 1$ by $- 1$ .

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - 25 x^{2} + 1 x^{4}} = 0$

Step 6.3.1.16.1.7

Multiply $x^{4}$ by $1$ .

$3 | 25 - x^{2} | \sqrt[3]{625 - 25 x^{2} - 25 x^{2} + x^{4}} = 0$

Step 6.3.1.16.2

Subtract $25 x^{2}$ from $- 25 x^{2}$ .

$3 | 25 - x^{2} | \sqrt[3]{625 - 50 x^{2} + x^{4}} = 0$

Step 6.3.1.17

Factor using the perfect square rule.

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Step 6.3.1.17.1

Rewrite $625$ as $25^{2}$ .

$3 | 25 - x^{2} | \sqrt[3]{25^{2} - 50 x^{2} + x^{4}} = 0$

Step 6.3.1.17.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$3 | 25 - x^{2} | \sqrt[3]{25^{2} - 50 x^{2} + {(x^{2})}^{2}} = 0$

Step 6.3.1.17.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$50 x^{2} = 2 \cdot 25 \cdot x^{2}$

Step 6.3.1.17.4

Rewrite the polynomial.

$3 | 25 - x^{2} | \sqrt[3]{25^{2} - 2 \cdot 25 \cdot x^{2} + {(x^{2})}^{2}} = 0$

Step 6.3.1.17.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 25$ and $b = x^{2}$ .

$3 | 25 - x^{2} | \sqrt[3]{{(25 - x^{2})}^{2}} = 0$

Step 6.3.1.18

Rewrite $25$ as $5^{2}$ .

$3 | 25 - x^{2} | \sqrt[3]{{(5^{2} - x^{2})}^{2}} = 0$

Step 6.3.1.19

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$3 | 25 - x^{2} | \sqrt[3]{{((5 + x) (5 - x))}^{2}} = 0$

Step 6.3.1.20

Apply the product rule to $(5 + x) (5 - x)$ .

$3 | 25 - x^{2} | \sqrt[3]{{(5 + x)}^{2} {(5 - x)}^{2}} = 0$

Step 6.3.1.21

Rewrite $25$ as $5^{2}$ .

$3 | 5^{2} - x^{2} | \sqrt[3]{{(5 + x)}^{2} {(5 - x)}^{2}} = 0$

Step 6.3.1.22

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

$3 | (5 + x) (5 - x) | \sqrt[3]{{(5 + x)}^{2} {(5 - x)}^{2}} = 0$

Step 6.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(5 + x)}^{2} {(5 - x)}^{2}}$ as ${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}}$ .

$3 | (5 + x) (5 - x) | {({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}} = 0$

Step 6.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$| (5 + x) (5 - x) | = 0$

${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}} = 0$

Step 6.3.4

Set $| (5 + x) (5 - x) |$ equal to $0$ and solve for $x$ .

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Step 6.3.4.1

Set $| (5 + x) (5 - x) |$ equal to $0$ .

$| (5 + x) (5 - x) | = 0$

Step 6.3.4.2

Solve $| (5 + x) (5 - x) | = 0$ for $x$ .

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Step 6.3.4.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$(5 + x) (5 - x) = \pm 0$

Step 6.3.4.2.2

Plus or minus $0$ is $0$ .

$(5 + x) (5 - x) = 0$

Step 6.3.4.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5 + x = 0$

$5 - x = 0$

Step 6.3.4.2.4

Set $5 + x$ equal to $0$ and solve for $x$ .

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Step 6.3.4.2.4.1

Set $5 + x$ equal to $0$ .

$5 + x = 0$

Step 6.3.4.2.4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6.3.4.2.5

Set $5 - x$ equal to $0$ and solve for $x$ .

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Step 6.3.4.2.5.1

Set $5 - x$ equal to $0$ .

$5 - x = 0$

Step 6.3.4.2.5.2

Solve $5 - x = 0$ for $x$ .

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Step 6.3.4.2.5.2.1

Subtract $5$ from both sides of the equation.

$- x = - 5$

Step 6.3.4.2.5.2.2

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 6.3.4.2.5.2.2.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 6.3.4.2.5.2.2.2

Simplify the left side.

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Step 6.3.4.2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 6.3.4.2.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 6.3.4.2.5.2.2.3

Simplify the right side.

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Step 6.3.4.2.5.2.2.3.1

Divide $- 5$ by $- 1$ .

$x = 5$

Step 6.3.4.2.6

The final solution is all the values that make $(5 + x) (5 - x) = 0$ true.

$x = - 5, 5$

Step 6.3.5

Set ${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}}$ equal to $0$ and solve for $x$ .

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Step 6.3.5.1

Set ${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}}$ equal to $0$ .

${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}} = 0$

Step 6.3.5.2

Solve ${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}} = 0$ for $x$ .

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Step 6.3.5.2.1

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 6.3.5.2.2

Simplify the exponent.

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Step 6.3.5.2.2.1

Simplify the left side.

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Step 6.3.5.2.2.1.1

Simplify ${({({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}})}^{3}$ .

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Step 6.3.5.2.2.1.1.1

Multiply the exponents in ${({({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}})}^{3}$ .

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Step 6.3.5.2.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.3.5.2.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 6.3.5.2.2.1.1.1.2.1

Cancel the common factor.

${({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.3.5.2.2.1.1.1.2.2

Rewrite the expression.

${({(5 + x)}^{2} {(5 - x)}^{2})}^{1} = 0^{3}$

Step 6.3.5.2.2.1.1.2

Simplify.

${(5 + x)}^{2} {(5 - x)}^{2} = 0^{3}$

Step 6.3.5.2.2.2

Simplify the right side.

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Step 6.3.5.2.2.2.1

Raising $0$ to any positive power yields $0$ .

${(5 + x)}^{2} {(5 - x)}^{2} = 0$

Step 6.3.5.2.3

Solve for $x$ .

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Step 6.3.5.2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(5 + x)}^{2} = 0$

${(5 - x)}^{2} = 0$

Step 6.3.5.2.3.2

Set ${(5 + x)}^{2}$ equal to $0$ and solve for $x$ .

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Step 6.3.5.2.3.2.1

Set ${(5 + x)}^{2}$ equal to $0$ .

${(5 + x)}^{2} = 0$

Step 6.3.5.2.3.2.2

Solve ${(5 + x)}^{2} = 0$ for $x$ .

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Step 6.3.5.2.3.2.2.1

Set the $5 + x$ equal to $0$ .

$5 + x = 0$

Step 6.3.5.2.3.2.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6.3.5.2.3.3

Set ${(5 - x)}^{2}$ equal to $0$ and solve for $x$ .

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Step 6.3.5.2.3.3.1

Set ${(5 - x)}^{2}$ equal to $0$ .

${(5 - x)}^{2} = 0$

Step 6.3.5.2.3.3.2

Solve ${(5 - x)}^{2} = 0$ for $x$ .

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Step 6.3.5.2.3.3.2.1

Set the $5 - x$ equal to $0$ .

$5 - x = 0$

Step 6.3.5.2.3.3.2.2

Solve for $x$ .

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Step 6.3.5.2.3.3.2.2.1

Subtract $5$ from both sides of the equation.

$- x = - 5$

Step 6.3.5.2.3.3.2.2.2

Divide each term in $- x = - 5$ by $- 1$ and simplify.

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Step 6.3.5.2.3.3.2.2.2.1

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Step 6.3.5.2.3.3.2.2.2.2

Simplify the left side.

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Step 6.3.5.2.3.3.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Step 6.3.5.2.3.3.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Step 6.3.5.2.3.3.2.2.2.3

Simplify the right side.

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Step 6.3.5.2.3.3.2.2.2.3.1

Divide $- 5$ by $- 1$ .

$x = 5$

Step 6.3.5.2.3.4

The final solution is all the values that make ${(5 + x)}^{2} {(5 - x)}^{2} = 0$ true.

$x = - 5, 5$

Step 6.3.6

The final solution is all the values that make $3 | (5 + x) (5 - x) | {({(5 + x)}^{2} {(5 - x)}^{2})}^{\frac{1}{3}} = 0$ true.

$x = - 5, 5$

Step 6.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 5, x = 5$

Step 7

Critical points to evaluate.

$x = 0, - 5, 5$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 {((0) + 5)}^{2} ({(0)}^{2} + 75) {((0) - 5)}^{2}}{9 {| 25 - {(0)}^{2} |}^{\frac{11}{3}}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Rewrite $0$ as $- 1 (0)$ .

$- \frac{2 {(- 1 (0) + 5)}^{2} (0^{2} + 75) {(0 - 5)}^{2}}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.2

Rewrite $5$ as $- 1 (- 5)$ .

$- \frac{2 {(- 1 \cdot 0 - 1 \cdot - 5)}^{2} (0^{2} + 75) {(0 - 5)}^{2}}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 5$ .

$- \frac{2 {(- 1 \cdot (0 - 5))}^{2} (0^{2} + 75) {(0 - 5)}^{2}}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.4

Apply the product rule to $- 1 \cdot (0 - 5)$ .

$- \frac{2 ({(- 1)}^{2} \cdot {(0 - 5)}^{2}) (0^{2} + 75) {(0 - 5)}^{2}}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.5

Raise $- 1$ to the power of $2$ .

$- \frac{2 (1 \cdot {(0 - 5)}^{2}) (0^{2} + 75) {(0 - 5)}^{2}}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.6

Multiply ${(0 - 5)}^{2}$ by $1$ .

$- \frac{2 {(0 - 5)}^{2} (0^{2} + 75) {(0 - 5)}^{2}}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.7

Multiply ${(0 - 5)}^{2}$ by ${(0 - 5)}^{2}$ by adding the exponents.

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Step 9.1.7.1

Move ${(0 - 5)}^{2}$ .

$- \frac{2 ({(0 - 5)}^{2} {(0 - 5)}^{2}) (0^{2} + 75)}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{2 {(0 - 5)}^{2 + 2} (0^{2} + 75)}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.1.7.3

Add $2$ and $2$ .

$- \frac{2 {(0 - 5)}^{4} (0^{2} + 75)}{9 {| 25 - 0^{2} |}^{\frac{11}{3}}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Raising $0$ to any positive power yields $0$ .

$- \frac{2 {(0 - 5)}^{4} (0^{2} + 75)}{9 {| 25 - 0 |}^{\frac{11}{3}}}$

Step 9.2.1.2

Multiply $- 1$ by $0$ .

$- \frac{2 {(0 - 5)}^{4} (0^{2} + 75)}{9 {| 25 + 0 |}^{\frac{11}{3}}}$

Step 9.2.2

Add $25$ and $0$ .

$- \frac{2 {(0 - 5)}^{4} (0^{2} + 75)}{9 {| 25 |}^{\frac{11}{3}}}$

Step 9.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $25$ is $25$ .

$- \frac{2 {(0 - 5)}^{4} (0^{2} + 75)}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

Subtract $5$ from $0$ .

$- \frac{2 {(- 5)}^{4} (0^{2} + 75)}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.3.2

Raising $0$ to any positive power yields $0$ .

$- \frac{2 {(- 5)}^{4} (0 + 75)}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.3.3

Add $0$ and $75$ .

$- \frac{2 {(- 5)}^{4} \cdot 75}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.3.4

Multiply $75$ by $2$ .

$- \frac{150 {(- 5)}^{4}}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.3.5

Raise $- 5$ to the power of $4$ .

$- \frac{150 \cdot 625}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.4

Simplify with factoring out.

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Step 9.4.1

Multiply $150$ by $625$ .

$- \frac{93750}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.4.2

Factor $3$ out of $93750$ .

$- \frac{3 (31250)}{9 \cdot 25^{\frac{11}{3}}}$

Step 9.5

Cancel the common factors.

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Step 9.5.1

Factor $3$ out of $9 \cdot 25^{\frac{11}{3}}$ .

$- \frac{3 (31250)}{3 (3 \cdot 25^{\frac{11}{3}})}$

Step 9.5.2

Cancel the common factor.

$- \frac{3 \cdot 31250}{3 (3 \cdot 25^{\frac{11}{3}})}$

Step 9.5.3

Rewrite the expression.

$- \frac{31250}{3 \cdot 25^{\frac{11}{3}}}$

Step 10

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \sqrt[3]{| 25 - {(0)}^{2} |} + 6$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \sqrt[3]{| 25 - 0 |} + 6$

Step 11.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = \sqrt[3]{| 25 + 0 |} + 6$

Step 11.2.1.3

Add $25$ and $0$ .

$f (0) = \sqrt[3]{| 25 |} + 6$

Step 11.2.1.4

The absolute value is the distance between a number and zero. The distance between $0$ and $25$ is $25$ .

$f (0) = \sqrt[3]{25} + 6$

Step 11.2.2

The final answer is $\sqrt[3]{25} + 6$ .

$y = \sqrt[3]{25} + 6$

Step 12

Evaluate the second derivative at $x = - 5$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 {| 25 - {(- 5)}^{2} |}^{\frac{11}{3}}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Raise $- 5$ to the power of $2$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 {| 25 - 1 \cdot 25 |}^{\frac{11}{3}}}$

Step 13.1.2

Multiply $- 1$ by $25$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 {| 25 - 25 |}^{\frac{11}{3}}}$

Step 13.2

Subtract $25$ from $25$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 {| 0 |}^{\frac{11}{3}}}$

Step 13.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 \cdot 0^{\frac{11}{3}}}$

Step 13.4

Simplify the expression.

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Step 13.4.1

Rewrite $0$ as $0^{3}$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 \cdot {(0^{3})}^{\frac{11}{3}}}$

Step 13.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 \cdot 0^{3 (\frac{11}{3})}}$

Step 13.5

Cancel the common factor of $3$ .

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Step 13.5.1

Cancel the common factor.

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 \cdot 0^{3 (\frac{11}{3})}}$

Step 13.5.2

Rewrite the expression.

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 \cdot 0^{11}}$

Step 13.6

Simplify the expression.

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Step 13.6.1

Raising $0$ to any positive power yields $0$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{9 \cdot 0}$

Step 13.6.2

Multiply $9$ by $0$ .

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{0}$

Step 13.6.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$- \frac{2 {((- 5) + 5)}^{2} ({(- 5)}^{2} + 75) {((- 5) - 5)}^{2}}{0}$

Step 13.7

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 14

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 14.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 5) \cup (- 5, 0) \cup (0, 5) \cup (5, \infty)$

Step 14.2

Substitute any number, such as $- 8$ , from the interval $(- \infty, - 5)$ in the first derivative $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ to check if the result is negative or positive.

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Step 14.2.1

Replace the variable $x$ with $- 8$ in the expression.

$f' (- 8) = - \frac{2 (- 8) (5 - 8) (5 - (- 8))}{3 {| 25 - {(- 8)}^{2} |}^{\frac{5}{3}}}$

Step 14.2.2

Simplify the result.

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Step 14.2.2.1

Simplify the expression.

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Step 14.2.2.1.1

Remove parentheses.

$f' (- 8) = - \frac{2 (- 8) (5 - 8) (5 - (- 8))}{3 {| 25 - {(- 8)}^{2} |}^{\frac{5}{3}}}$

Step 14.2.2.1.2

Multiply $2$ by $- 8$ .

$f' (- 8) = - \frac{- 16 (5 - 8) (5 - (- 8))}{3 {| 25 - {(- 8)}^{2} |}^{\frac{5}{3}}}$

Step 14.2.2.2

Simplify the denominator.

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Step 14.2.2.2.1

Simplify each term.

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Step 14.2.2.2.1.1

Raise $- 8$ to the power of $2$ .

$f' (- 8) = - \frac{- 16 (5 - 8) (5 - (- 8))}{3 {| 25 - 1 \cdot 64 |}^{\frac{5}{3}}}$

Step 14.2.2.2.1.2

Multiply $- 1$ by $64$ .

$f' (- 8) = - \frac{- 16 (5 - 8) (5 - (- 8))}{3 {| 25 - 64 |}^{\frac{5}{3}}}$

Step 14.2.2.2.2

Subtract $64$ from $25$ .

$f' (- 8) = - \frac{- 16 (5 - 8) (5 - (- 8))}{3 {| - 39 |}^{\frac{5}{3}}}$

Step 14.2.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 39$ and $0$ is $39$ .

$f' (- 8) = - \frac{- 16 (5 - 8) (5 - (- 8))}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.3

Simplify the numerator.

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Step 14.2.2.3.1

Multiply $- 1$ by $- 8$ .

$f' (- 8) = - \frac{- 16 (5 - 8) (5 + 8)}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.3.2

Subtract $8$ from $5$ .

$f' (- 8) = - \frac{- 16 \cdot (- 3 (5 + 8))}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.3.3

Multiply $- 16$ by $- 3$ .

$f' (- 8) = - \frac{48 (5 + 8)}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.3.4

Add $5$ and $8$ .

$f' (- 8) = - \frac{48 \cdot 13}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.4

Simplify with factoring out.

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Step 14.2.2.4.1

Multiply $48$ by $13$ .

$f' (- 8) = - \frac{624}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.4.2

Factor $3$ out of $624$ .

$f' (- 8) = - \frac{3 \cdot 208}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.5

Cancel the common factors.

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Step 14.2.2.5.1

Factor $3$ out of $3 \cdot 39^{\frac{5}{3}}$ .

$f' (- 8) = - \frac{3 \cdot 208}{3 (39^{\frac{5}{3}})}$

Step 14.2.2.5.2

Cancel the common factor.

$f' (- 8) = - \frac{3 \cdot 208}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.2.2.5.3

Rewrite the expression.

$f' (- 8) = - \frac{208}{39^{\frac{5}{3}}}$

Step 14.2.2.6

The final answer is $- \frac{208}{39^{\frac{5}{3}}}$ .

$- \frac{208}{39^{\frac{5}{3}}}$

Step 14.3

Substitute any number, such as $- 2$ , from the interval $(- 5, 0)$ in the first derivative $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ to check if the result is negative or positive.

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Step 14.3.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = - \frac{2 (- 2) (5 - 2) (5 - (- 2))}{3 {| 25 - {(- 2)}^{2} |}^{\frac{5}{3}}}$

Step 14.3.2

Simplify the result.

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Step 14.3.2.1

Simplify the expression.

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Step 14.3.2.1.1

Remove parentheses.

$f' (- 2) = - \frac{2 (- 2) (5 - 2) (5 - (- 2))}{3 {| 25 - {(- 2)}^{2} |}^{\frac{5}{3}}}$

Step 14.3.2.1.2

Multiply $2$ by $- 2$ .

$f' (- 2) = - \frac{- 4 (5 - 2) (5 - (- 2))}{3 {| 25 - {(- 2)}^{2} |}^{\frac{5}{3}}}$

Step 14.3.2.2

Simplify the denominator.

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Step 14.3.2.2.1

Simplify each term.

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Step 14.3.2.2.1.1

Raise $- 2$ to the power of $2$ .

$f' (- 2) = - \frac{- 4 (5 - 2) (5 - (- 2))}{3 {| 25 - 1 \cdot 4 |}^{\frac{5}{3}}}$

Step 14.3.2.2.1.2

Multiply $- 1$ by $4$ .

$f' (- 2) = - \frac{- 4 (5 - 2) (5 - (- 2))}{3 {| 25 - 4 |}^{\frac{5}{3}}}$

Step 14.3.2.2.2

Subtract $4$ from $25$ .

$f' (- 2) = - \frac{- 4 (5 - 2) (5 - (- 2))}{3 {| 21 |}^{\frac{5}{3}}}$

Step 14.3.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $21$ is $21$ .

$f' (- 2) = - \frac{- 4 (5 - 2) (5 - (- 2))}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.3

Simplify the numerator.

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Step 14.3.2.3.1

Multiply $- 1$ by $- 2$ .

$f' (- 2) = - \frac{- 4 (5 - 2) (5 + 2)}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.3.2

Subtract $2$ from $5$ .

$f' (- 2) = - \frac{- 4 \cdot (3 (5 + 2))}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.3.3

Multiply $- 4$ by $3$ .

$f' (- 2) = - \frac{- 12 (5 + 2)}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.3.4

Add $5$ and $2$ .

$f' (- 2) = - \frac{- 12 \cdot 7}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.4

Simplify with factoring out.

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Step 14.3.2.4.1

Multiply $- 12$ by $7$ .

$f' (- 2) = - \frac{- 84}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.4.2

Factor $3$ out of $- 84$ .

$f' (- 2) = - \frac{3 \cdot - 28}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.5

Cancel the common factors.

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Step 14.3.2.5.1

Factor $3$ out of $3 \cdot 21^{\frac{5}{3}}$ .

$f' (- 2) = - \frac{3 \cdot - 28}{3 (21^{\frac{5}{3}})}$

Step 14.3.2.5.2

Cancel the common factor.

$f' (- 2) = - \frac{3 \cdot - 28}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.3.2.5.3

Rewrite the expression.

$f' (- 2) = - \frac{- 28}{21^{\frac{5}{3}}}$

Step 14.3.2.6

Move the negative in front of the fraction.

$f' (- 2) = \frac{28}{21^{\frac{5}{3}}}$

Step 14.3.2.7

The final answer is $\frac{28}{21^{\frac{5}{3}}}$ .

$\frac{28}{21^{\frac{5}{3}}}$

Step 14.4

Substitute any number, such as $2$ , from the interval $(0, 5)$ in the first derivative $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ to check if the result is negative or positive.

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Step 14.4.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = - \frac{2 (2) (5 + 2) (5 - (2))}{3 {| 25 - {(2)}^{2} |}^{\frac{5}{3}}}$

Step 14.4.2

Simplify the result.

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Step 14.4.2.1

Simplify the expression.

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Step 14.4.2.1.1

Remove parentheses.

$f' (2) = - \frac{2 (2) (5 + 2) (5 - (2))}{3 {| 25 - {(2)}^{2} |}^{\frac{5}{3}}}$

Step 14.4.2.1.2

Multiply $2$ by $2$ .

$f' (2) = - \frac{4 (5 + 2) (5 - (2))}{3 {| 25 - 2^{2} |}^{\frac{5}{3}}}$

Step 14.4.2.2

Simplify the denominator.

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Step 14.4.2.2.1

Simplify each term.

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Step 14.4.2.2.1.1

Raise $2$ to the power of $2$ .

$f' (2) = - \frac{4 (5 + 2) (5 - (2))}{3 {| 25 - 1 \cdot 4 |}^{\frac{5}{3}}}$

Step 14.4.2.2.1.2

Multiply $- 1$ by $4$ .

$f' (2) = - \frac{4 (5 + 2) (5 - (2))}{3 {| 25 - 4 |}^{\frac{5}{3}}}$

Step 14.4.2.2.2

Subtract $4$ from $25$ .

$f' (2) = - \frac{4 (5 + 2) (5 - (2))}{3 {| 21 |}^{\frac{5}{3}}}$

Step 14.4.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $21$ is $21$ .

$f' (2) = - \frac{4 (5 + 2) (5 - (2))}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.3

Simplify the numerator.

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Step 14.4.2.3.1

Multiply $- 1$ by $2$ .

$f' (2) = - \frac{4 (5 + 2) (5 - 2)}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.3.2

Add $5$ and $2$ .

$f' (2) = - \frac{4 \cdot (7 (5 - 2))}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.3.3

Multiply $4$ by $7$ .

$f' (2) = - \frac{28 (5 - 2)}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.3.4

Subtract $2$ from $5$ .

$f' (2) = - \frac{28 \cdot 3}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.4

Simplify with factoring out.

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Step 14.4.2.4.1

Multiply $28$ by $3$ .

$f' (2) = - \frac{84}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.4.2

Factor $3$ out of $84$ .

$f' (2) = - \frac{3 \cdot 28}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.5

Cancel the common factors.

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Step 14.4.2.5.1

Factor $3$ out of $3 \cdot 21^{\frac{5}{3}}$ .

$f' (2) = - \frac{3 \cdot 28}{3 (21^{\frac{5}{3}})}$

Step 14.4.2.5.2

Cancel the common factor.

$f' (2) = - \frac{3 \cdot 28}{3 \cdot 21^{\frac{5}{3}}}$

Step 14.4.2.5.3

Rewrite the expression.

$f' (2) = - \frac{28}{21^{\frac{5}{3}}}$

Step 14.4.2.6

The final answer is $- \frac{28}{21^{\frac{5}{3}}}$ .

$- \frac{28}{21^{\frac{5}{3}}}$

Step 14.5

Substitute any number, such as $8$ , from the interval $(5, \infty)$ in the first derivative $- \frac{2 x (5 + x) (5 - x)}{3 {| 25 - x^{2} |}^{\frac{5}{3}}}$ to check if the result is negative or positive.

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Step 14.5.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = - \frac{2 (8) (5 + 8) (5 - (8))}{3 {| 25 - {(8)}^{2} |}^{\frac{5}{3}}}$

Step 14.5.2

Simplify the result.

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Step 14.5.2.1

Simplify the expression.

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Step 14.5.2.1.1

Remove parentheses.

$f' (8) = - \frac{2 (8) (5 + 8) (5 - (8))}{3 {| 25 - {(8)}^{2} |}^{\frac{5}{3}}}$

Step 14.5.2.1.2

Multiply $2$ by $8$ .

$f' (8) = - \frac{16 (5 + 8) (5 - (8))}{3 {| 25 - 8^{2} |}^{\frac{5}{3}}}$

Step 14.5.2.2

Simplify the denominator.

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Step 14.5.2.2.1

Simplify each term.

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Step 14.5.2.2.1.1

Raise $8$ to the power of $2$ .

$f' (8) = - \frac{16 (5 + 8) (5 - (8))}{3 {| 25 - 1 \cdot 64 |}^{\frac{5}{3}}}$

Step 14.5.2.2.1.2

Multiply $- 1$ by $64$ .

$f' (8) = - \frac{16 (5 + 8) (5 - (8))}{3 {| 25 - 64 |}^{\frac{5}{3}}}$

Step 14.5.2.2.2

Subtract $64$ from $25$ .

$f' (8) = - \frac{16 (5 + 8) (5 - (8))}{3 {| - 39 |}^{\frac{5}{3}}}$

Step 14.5.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 39$ and $0$ is $39$ .

$f' (8) = - \frac{16 (5 + 8) (5 - (8))}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.3

Simplify the numerator.

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Step 14.5.2.3.1

Multiply $- 1$ by $8$ .

$f' (8) = - \frac{16 (5 + 8) (5 - 8)}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.3.2

Add $5$ and $8$ .

$f' (8) = - \frac{16 \cdot (13 (5 - 8))}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.3.3

Multiply $16$ by $13$ .

$f' (8) = - \frac{208 (5 - 8)}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.3.4

Subtract $8$ from $5$ .

$f' (8) = - \frac{208 \cdot - 3}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.4

Simplify with factoring out.

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Step 14.5.2.4.1

Multiply $208$ by $- 3$ .

$f' (8) = - \frac{- 624}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.4.2

Factor $3$ out of $- 624$ .

$f' (8) = - \frac{3 \cdot - 208}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.5

Cancel the common factors.

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Step 14.5.2.5.1

Factor $3$ out of $3 \cdot 39^{\frac{5}{3}}$ .

$f' (8) = - \frac{3 \cdot - 208}{3 (39^{\frac{5}{3}})}$

Step 14.5.2.5.2

Cancel the common factor.

$f' (8) = - \frac{3 \cdot - 208}{3 \cdot 39^{\frac{5}{3}}}$

Step 14.5.2.5.3

Rewrite the expression.

$f' (8) = - \frac{- 208}{39^{\frac{5}{3}}}$

Step 14.5.2.6

Move the negative in front of the fraction.

$f' (8) = \frac{208}{39^{\frac{5}{3}}}$

Step 14.5.2.7

The final answer is $\frac{208}{39^{\frac{5}{3}}}$ .

$\frac{208}{39^{\frac{5}{3}}}$

Step 14.6

Since the first derivative changed signs from negative to positive around $x = - 5$ , then $x = - 5$ is a local minimum.

$x = - 5$ is a local minimum

Step 14.7

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 14.8

Since the first derivative changed signs from negative to positive around $x = 5$ , then $x = 5$ is a local minimum.

$x = 5$ is a local minimum

Step 14.9

These are the local extrema for $f (x) = \sqrt[3]{| 25 - x^{2} |} + 6$ .

$x = - 5$ is a local minimum

$x = 0$ is a local maximum

$x = 5$ is a local minimum

$x = - 5$ is a local minimum

$x = 0$ is a local maximum

$x = 5$ is a local minimum

Step 15