Calculus Examples

$f (x) = \sqrt[3]{5 x^{2} + 15}$

Step 1

Find the first derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 x^{2} + 15}$ as ${(5 x^{2} + 15)}^{\frac{1}{3}}$ .

$f' (x) = \frac{d}{d x} ({(5 x^{2} + 15)}^{\frac{1}{3}})$

Step 1.2

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 $g (x) = 5 x^{2} + 15$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $5 x^{2} + 15$ .

$f' (x) = \frac{d}{d u} (u^{\frac{1}{3}}) \frac{d}{d x} (5 x^{2} + 15)$

Step 1.2.2

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

$f' (x) = \frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.2.3

Replace all occurrences of $u$ with $5 x^{2} + 15$ .

$f' (x) = \frac{1}{3} \cdot {(5 x^{2} + 15)}^{\frac{1}{3} - 1} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.3

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

$f' (x) = \frac{1}{3} \cdot {(5 x^{2} + 15)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.4

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

$f' (x) = \frac{1}{3} \cdot {(5 x^{2} + 15)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.5

Combine the numerators over the common denominator.

$f' (x) = \frac{1}{3} \cdot {(5 x^{2} + 15)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $3$ .

$f' (x) = \frac{1}{3} \cdot {(5 x^{2} + 15)}^{\frac{1 - 3}{3}} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.6.2

Subtract $3$ from $1$ .

$f' (x) = \frac{1}{3} \cdot {(5 x^{2} + 15)}^{\frac{- 2}{3}} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

$f' (x) = \frac{1}{3} \cdot {(5 x^{2} + 15)}^{- \frac{2}{3}} \frac{d}{d x} (5 x^{2} + 15)$

Step 1.7.2

Combine $\frac{1}{3}$ and ${(5 x^{2} + 15)}^{- \frac{2}{3}}$ .

$f' (x) = \frac{{(5 x^{2} + 15)}^{- \frac{2}{3}}}{3} \cdot \frac{d}{d x} (5 x^{2} + 15)$

Step 1.7.3

Move ${(5 x^{2} + 15)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot \frac{d}{d x} (5 x^{2} + 15)$

Step 1.8

By the Sum Rule, the derivative of $5 x^{2} + 15$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [15]$ .

$f' (x) = \frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot (\frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (15))$

Step 1.9

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$f' (x) = \frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot (5 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (15))$

Step 1.10

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

$f' (x) = \frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot (5 (2 x) + \frac{d}{d x} (15))$

Step 1.11

Multiply $2$ by $5$ .

$f' (x) = \frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot (10 x + \frac{d}{d x} (15))$

Step 1.12

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$f' (x) = \frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot (10 x + 0)$

Step 1.13

Combine fractions.

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Step 1.13.1

Add $10 x$ and $0$ .

$f' (x) = \frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot (10 x)$

Step 1.13.2

Combine $10$ and $\frac{1}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}}$ .

$f' (x) = \frac{10}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}} \cdot x$

Step 1.13.3

Combine $\frac{10}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}}$ and $x$ .

$f' (x) = \frac{10 x}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{10}{3}$ is constant with respect to $x$ , the derivative of $\frac{10 x}{3 {(5 x^{2} + 15)}^{\frac{2}{3}}}$ with respect to $x$ is $\frac{10}{3} \frac{d}{d x} [\frac{x}{{(5 x^{2} + 15)}^{\frac{2}{3}}}]$ .

$f'' (x) = \frac{10}{3} \cdot \frac{d}{d x} (\frac{x}{{(5 x^{2} + 15)}^{\frac{2}{3}}})$

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 $f (x) = x$ and $g (x) = {(5 x^{2} + 15)}^{\frac{2}{3}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} \frac{d}{d x} (x) - x \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{2}{3}}}{{({(5 x^{2} + 15)}^{\frac{2}{3}})}^{2}}$

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

Multiply the exponents in ${({(5 x^{2} + 15)}^{\frac{2}{3}})}^{2}$ .

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Step 2.3.1.1

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} \frac{d}{d x} (x) - x \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{2}{3}}}{{(5 x^{2} + 15)}^{\frac{2}{3} \cdot 2}}$

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} \frac{d}{d x} (x) - x \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{2}{3}}}{{(5 x^{2} + 15)}^{\frac{2 \cdot 2}{3}}}$

Step 2.3.1.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} \frac{d}{d x} (x) - x \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{2}{3}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.3.2

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} \cdot 1 - x \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{2}{3}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.3.3

Multiply ${(5 x^{2} + 15)}^{\frac{2}{3}}$ by $1$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{2}{3}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 5 x^{2} + 15$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $5 x^{2} + 15$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{d}{d u} (u^{\frac{2}{3}}) \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.4.3

Replace all occurrences of $u$ with $5 x^{2} + 15$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} \cdot {(5 x^{2} + 15)}^{\frac{2}{3} - 1} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.5

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} \cdot {(5 x^{2} + 15)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.6

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} \cdot {(5 x^{2} + 15)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.7

Combine the numerators over the common denominator.

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} \cdot {(5 x^{2} + 15)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Multiply $- 1$ by $3$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} \cdot {(5 x^{2} + 15)}^{\frac{2 - 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.8.2

Subtract $3$ from $2$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} \cdot {(5 x^{2} + 15)}^{\frac{- 1}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.9

Combine fractions.

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Step 2.9.1

Move the negative in front of the fraction.

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3} \cdot {(5 x^{2} + 15)}^{- \frac{1}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.9.2

Combine $\frac{2}{3}$ and ${(5 x^{2} + 15)}^{- \frac{1}{3}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2 {(5 x^{2} + 15)}^{- \frac{1}{3}}}{3} \cdot \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.9.3

Move ${(5 x^{2} + 15)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - x (\frac{2}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.9.4

Combine $\frac{2}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}$ and $x$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot \frac{d}{d x} (5 x^{2} + 15)}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.10

By the Sum Rule, the derivative of $5 x^{2} + 15$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [15]$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot (\frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.11

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot (5 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.12

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot (5 (2 x) + \frac{d}{d x} (15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.13

Multiply $2$ by $5$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot (10 x + \frac{d}{d x} (15))}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.14

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot (10 x + 0)}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.15

Combine fractions.

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Step 2.15.1

Add $10 x$ and $0$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot (10 x)}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.15.2

Multiply $10$ by $- 1$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - 10 \frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot x}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.15.3

Combine $- 10$ and $\frac{2 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} + \frac{- 10 (2 x)}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot x}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.15.4

Multiply $2$ by $- 10$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} + \frac{- 20 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot x}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.15.5

Combine $\frac{- 20 x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}$ and $x$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} + \frac{- 20 x \cdot x}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.16

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} + \frac{- 20 (x \cdot x)}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.17

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} + \frac{- 20 (x \cdot x)}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.18

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

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} + \frac{- 20 x^{1 + 1}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.19

Add $1$ and $1$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} + \frac{- 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.20

Move the negative in front of the fraction.

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} - \frac{(20) x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.21

To write ${(5 x^{2} + 15)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{{(5 x^{2} + 15)}^{\frac{2}{3}} \cdot \frac{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} - \frac{20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.22

Combine ${(5 x^{2} + 15)}^{\frac{2}{3}}$ and $\frac{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{{(5 x^{2} + 15)}^{\frac{2}{3}} (3 {(5 x^{2} + 15)}^{\frac{1}{3}})}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} - \frac{20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.23

Combine the numerators over the common denominator.

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{{(5 x^{2} + 15)}^{\frac{2}{3}} (3 {(5 x^{2} + 15)}^{\frac{1}{3}}) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.24

Multiply ${(5 x^{2} + 15)}^{\frac{2}{3}}$ by ${(5 x^{2} + 15)}^{\frac{1}{3}}$ by adding the exponents.

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Step 2.24.1

Move ${(5 x^{2} + 15)}^{\frac{1}{3}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{{(5 x^{2} + 15)}^{\frac{1}{3}} {(5 x^{2} + 15)}^{\frac{2}{3}} \cdot 3 - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.24.2

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

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{{(5 x^{2} + 15)}^{\frac{1}{3} + \frac{2}{3}} \cdot 3 - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.24.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{{(5 x^{2} + 15)}^{\frac{1 + 2}{3}} \cdot 3 - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.24.4

Add $1$ and $2$ .

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{{(5 x^{2} + 15)}^{\frac{3}{3}} \cdot 3 - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.24.5

Divide $3$ by $3$ .

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{(5 x^{2} + 15) \cdot 3 - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.25

Simplify ${(5 x^{2} + 15)}^{1} \cdot 3$ .

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{(5 x^{2} + 15) \cdot 3 - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.26

Move $3$ to the left of $5 x^{2} + 15$ .

$f'' (x) = \frac{10}{3} \cdot \frac{\frac{3 \cdot (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.27

Rewrite $\frac{\frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$ as a product.

$f'' (x) = \frac{10}{3} \cdot (\frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}} \cdot \frac{1}{{(5 x^{2} + 15)}^{\frac{4}{3}}})$

Step 2.28

Multiply $\frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}}}$ by $\frac{1}{{(5 x^{2} + 15)}^{\frac{4}{3}}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{1}{3}} {(5 x^{2} + 15)}^{\frac{4}{3}}}$

Step 2.29

Multiply ${(5 x^{2} + 15)}^{\frac{1}{3}}$ by ${(5 x^{2} + 15)}^{\frac{4}{3}}$ by adding the exponents.

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Step 2.29.1

Move ${(5 x^{2} + 15)}^{\frac{4}{3}}$ .

$f'' (x) = \frac{10}{3} \cdot \frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 ({(5 x^{2} + 15)}^{\frac{4}{3}} {(5 x^{2} + 15)}^{\frac{1}{3}})}$

Step 2.29.2

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

$f'' (x) = \frac{10}{3} \cdot \frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{4}{3} + \frac{1}{3}}}$

Step 2.29.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{10}{3} \cdot \frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{4 + 1}{3}}}$

Step 2.29.4

Add $4$ and $1$ .

$f'' (x) = \frac{10}{3} \cdot \frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.30

Multiply $\frac{10}{3}$ by $\frac{3 (5 x^{2} + 15) - 20 x^{2}}{3 {(5 x^{2} + 15)}^{\frac{5}{3}}}$ .

$f'' (x) = \frac{10 (3 (5 x^{2} + 15) - 20 x^{2})}{3 (3 {(5 x^{2} + 15)}^{\frac{5}{3}})}$

Step 2.31

Multiply $3$ by $3$ .

$f'' (x) = \frac{10 (3 (5 x^{2} + 15) - 20 x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32

Simplify.

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Step 2.32.1

Apply the distributive property.

$f'' (x) = \frac{10 (3 (5 x^{2}) + 3 \cdot 15 - 20 x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.2

Apply the distributive property.

$f'' (x) = \frac{10 (3 (5 x^{2})) + 10 (3 \cdot 15) + 10 (- 20 x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.3

Simplify the numerator.

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Step 2.32.3.1

Simplify each term.

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Step 2.32.3.1.1

Multiply $5$ by $3$ .

$f'' (x) = \frac{10 (15 x^{2}) + 10 (3 \cdot 15) + 10 (- 20 x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.3.1.2

Multiply $15$ by $10$ .

$f'' (x) = \frac{150 x^{2} + 10 (3 \cdot 15) + 10 (- 20 x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.3.1.3

Multiply $10 (3 \cdot 15)$ .

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Step 2.32.3.1.3.1

Multiply $3$ by $15$ .

$f'' (x) = \frac{150 x^{2} + 10 \cdot 45 + 10 (- 20 x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.3.1.3.2

Multiply $10$ by $45$ .

$f'' (x) = \frac{150 x^{2} + 450 + 10 (- 20 x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.3.1.4

Multiply $- 20$ by $10$ .

$f'' (x) = \frac{150 x^{2} + 450 - 200 x^{2}}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.3.2

Subtract $200 x^{2}$ from $150 x^{2}$ .

$f'' (x) = \frac{- 50 x^{2} + 450}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.4

Simplify the numerator.

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Step 2.32.4.1

Factor $50$ out of $- 50 x^{2} + 450$ .

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Step 2.32.4.1.1

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

$f'' (x) = \frac{50 (- x^{2}) + 450}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.4.1.2

Factor $50$ out of $450$ .

$f'' (x) = \frac{50 (- x^{2}) + 50 (9)}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.4.1.3

Factor $50$ out of $50 (- x^{2}) + 50 (9)$ .

$f'' (x) = \frac{50 (- x^{2} + 9)}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.4.2

Rewrite $9$ as $3^{2}$ .

$f'' (x) = \frac{50 (- x^{2} + 3^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.4.3

Reorder $- x^{2}$ and $3^{2}$ .

$f'' (x) = \frac{50 (3^{2} - x^{2})}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 2.32.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$f'' (x) = \frac{50 (3 + x) (3 - x)}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$

Step 3

Find the third derivative.

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Step 3.1

Since $\frac{50}{9}$ is constant with respect to $x$ , the derivative of $\frac{50 (3 + x) (3 - x)}{9 {(5 x^{2} + 15)}^{\frac{5}{3}}}$ with respect to $x$ is $\frac{50}{9} \frac{d}{d x} [\frac{(3 + x) (3 - x)}{{(5 x^{2} + 15)}^{\frac{5}{3}}}]$ .

$f''' (x) = \frac{50}{9} \cdot \frac{d}{d x} (\frac{(3 + x) (3 - x)}{{(5 x^{2} + 15)}^{\frac{5}{3}}})$

Step 3.2

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} \frac{d}{d x} ((3 + x) (3 - x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{({(5 x^{2} + 15)}^{\frac{5}{3}})}^{2}}$

Step 3.3

Multiply the exponents in ${({(5 x^{2} + 15)}^{\frac{5}{3}})}^{2}$ .

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Step 3.3.1

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} \frac{d}{d x} ((3 + x) (3 - x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{5}{3} \cdot 2}}$

Step 3.3.2

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

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Step 3.3.2.1

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

Step 3.3.2.2

Multiply $5$ by $2$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} \frac{d}{d x} ((3 + x) (3 - x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.4

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 $f (x) = 3 + x$ and $g (x) = 3 - x$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} ((3 + x) \frac{d}{d x} (3 - x) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5

Differentiate.

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Step 3.5.1

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} ((3 + x) (\frac{d}{d x} (3) + \frac{d}{d x} (- x)) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} ((3 + x) (0 + \frac{d}{d x} (- x)) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.3

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} ((3 + x) \frac{d}{d x} (- x) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.4

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} ((3 + x) (- \frac{d}{d x} x) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.5

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} ((3 + x) (- 1 \cdot 1) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.6

Simplify the expression.

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Step 3.5.6.1

Multiply $- 1$ by $1$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} ((3 + x) \cdot - 1 + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.6.2

Move $- 1$ to the left of $3 + x$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot (3 + x) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.6.3

Rewrite $- 1 (3 + x)$ as $- (3 + x)$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + (3 - x) \frac{d}{d x} (3 + x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.7

By the Sum Rule, the derivative of $3 + x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [x]$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + (3 - x) (\frac{d}{d x} (3) + \frac{d}{d x} (x))) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.8

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + (3 - x) (0 + \frac{d}{d x} (x))) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.9

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + (3 - x) \frac{d}{d x} (x)) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.10

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + (3 - x) \cdot 1) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.5.11

Multiply $3 - x$ by $1$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{5}{3}}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.6

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 $g (x) = 5 x^{2} + 15$ .

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Step 3.6.1

To apply the Chain Rule, set $u$ as $5 x^{2} + 15$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{d}{d u} (u^{\frac{5}{3}}) \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.6.2

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5}{3} u^{\frac{5}{3} - 1} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.6.3

Replace all occurrences of $u$ with $5 x^{2} + 15$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5}{3} \cdot {(5 x^{2} + 15)}^{\frac{5}{3} - 1} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.7

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5}{3} \cdot {(5 x^{2} + 15)}^{\frac{5}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.8

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5}{3} \cdot {(5 x^{2} + 15)}^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.9

Combine the numerators over the common denominator.

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5}{3} \cdot {(5 x^{2} + 15)}^{\frac{5 - 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $3$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5}{3} \cdot {(5 x^{2} + 15)}^{\frac{5 - 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.10.2

Subtract $3$ from $5$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5}{3} \cdot {(5 x^{2} + 15)}^{\frac{2}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.11

Combine $\frac{5}{3}$ and ${(5 x^{2} + 15)}^{\frac{2}{3}}$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.12

By the Sum Rule, the derivative of $5 x^{2} + 15$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [15]$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot (\frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.13

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot (5 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.14

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

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot (5 (2 x) + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.15

Multiply $2$ by $5$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot (10 x + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.16

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot (10 x + 0))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.17

Combine fractions.

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Step 3.17.1

Add $10 x$ and $0$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot (10 x))}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.17.2

Combine $10$ and $\frac{5 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3}$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{10 (5 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3} \cdot x)}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.17.3

Multiply $5$ by $10$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3} \cdot x)}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.17.4

Combine $\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}}}{3}$ and $x$ .

$f''' (x) = \frac{50}{9} \cdot \frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.17.5

Multiply $\frac{50}{9}$ by $\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}}{{(5 x^{2} + 15)}^{\frac{10}{3}}}$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- (3 + x) + 3 - x) - (3 + x) (3 - x) \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18

Simplify.

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Step 3.18.1

Apply the distributive property.

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x) - (3 + x) (3 - x) \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.2

Apply the distributive property.

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x) + (- 1 \cdot 3 - x) (3 - x) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.3

Apply the distributive property.

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x)) + 50 ((- 1 \cdot 3 - x) (3 - x) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4

Simplify the numerator.

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Step 3.18.4.1

Factor $50$ out of $50 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x) + 50 (- 1 \cdot 3 - x) (3 - x) \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}$ .

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Step 3.18.4.1.1

Factor $50$ out of $50 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x)$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x)) + 50 (- 1 \cdot 3 - x) (3 - x) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.1.2

Factor $50$ out of $50 (- 1 \cdot 3 - x) (3 - x) \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x)) + 50 (((- 1 \cdot 3 - x) (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.1.3

Factor $50$ out of $50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x)) + 50 (((- 1 \cdot 3 - x) (3 - x)) \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3})$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 1 \cdot 3 - x + 3 - x) + ((- 1 \cdot 3 - x) (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.2

Multiply $- 1$ by $3$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 3 - x + 3 - x) + ((- 1 \cdot 3 - x) (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.3

Combine the opposite terms in $- 3 - x + 3 - x$ .

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Step 3.18.4.3.1

Add $- 3$ and $3$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- x + 0 - x) + ((- 1 \cdot 3 - x) (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.3.2

Add $- x$ and $0$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- x - x) + ((- 1 \cdot 3 - x) (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.4

Subtract $x$ from $- x$ .

$f''' (x) = \frac{50 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 2 x) + ((- 1 \cdot 3 - x) (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.5

Rewrite using the commutative property of multiplication.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + ((- 1 \cdot 3 - x) (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.6

Multiply $- 1$ by $3$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 3 - x) (3 - x) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.7

Expand $(- 3 - x) (3 - x)$ using the FOIL Method.

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Step 3.18.4.7.1

Apply the distributive property.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 3 (3 - x) - x (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.7.2

Apply the distributive property.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 3 \cdot 3 - 3 (- x) - x (3 - x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.7.3

Apply the distributive property.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 3 \cdot 3 - 3 (- x) - x \cdot 3 - x (- x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8

Simplify and combine like terms.

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Step 3.18.4.8.1

Simplify each term.

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Step 3.18.4.8.1.1

Multiply $- 3$ by $3$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 - 3 (- x) - x \cdot 3 - x (- x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.1.2

Multiply $- 1$ by $- 3$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 3 x - x \cdot 3 - x (- x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.1.3

Multiply $3$ by $- 1$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 3 x - 3 x - x (- x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.1.4

Rewrite using the commutative property of multiplication.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 3 x - 3 x - 1 \cdot (- 1 x \cdot x)) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.1.5

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

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Step 3.18.4.8.1.5.1

Move $x$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 3 x - 3 x - 1 \cdot (- 1 (x \cdot x))) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.1.5.2

Multiply $x$ by $x$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 3 x - 3 x - 1 \cdot (- 1 x^{2})) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.1.6

Multiply $- 1$ by $- 1$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 3 x - 3 x + 1 x^{2}) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.1.7

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

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 3 x - 3 x + x^{2}) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.2

Subtract $3 x$ from $3 x$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + 0 + x^{2}) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.8.3

Add $- 9$ and $0$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + (- 9 + x^{2}) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.9

Apply the distributive property.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 9 \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3} + x^{2} (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.10

Cancel the common factor of $3$ .

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Step 3.18.4.10.1

Factor $3$ out of $- 9$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + 3 (- 3) (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}) + x^{2} (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.10.2

Cancel the common factor.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + 3 \cdot (- 3 \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}) + x^{2} (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.10.3

Rewrite the expression.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 3 (50 {(5 x^{2} + 15)}^{\frac{2}{3}} x) + x^{2} (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.11

Multiply $50$ by $- 3$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 ({(5 x^{2} + 15)}^{\frac{2}{3}} x) + x^{2} (\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}))}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.12

Multiply $x^{2} \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}$ .

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Step 3.18.4.12.1

Combine $x^{2}$ and $\frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x}{3}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 ({(5 x^{2} + 15)}^{\frac{2}{3}} x) + \frac{x^{2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}} x)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.12.2

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

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Step 3.18.4.12.2.1

Move $x$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 ({(5 x^{2} + 15)}^{\frac{2}{3}} x) + \frac{x \cdot x^{2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.12.2.2

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

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Step 3.18.4.12.2.2.1

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

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 ({(5 x^{2} + 15)}^{\frac{2}{3}} x) + \frac{x \cdot x^{2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.12.2.2.2

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

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 ({(5 x^{2} + 15)}^{\frac{2}{3}} x) + \frac{x^{1 + 2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.12.2.3

Add $1$ and $2$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 ({(5 x^{2} + 15)}^{\frac{2}{3}} x) + \frac{x^{3} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.13

Simplify each term.

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Step 3.18.4.13.1

Simplify the numerator.

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Step 3.18.4.13.1.1

Rewrite.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x + \frac{x^{2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}} x)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.13.1.2

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

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Step 3.18.4.13.1.2.1

Move $x$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x + \frac{x \cdot x^{2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.13.1.2.2

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

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Step 3.18.4.13.1.2.2.1

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

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x + \frac{x \cdot x^{2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.13.1.2.2.2

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

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x + \frac{x^{1 + 2} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.13.1.2.3

Add $1$ and $2$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x + \frac{x^{3} (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.13.1.3

Remove unnecessary parentheses.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x + \frac{x^{3} \cdot (50 {(5 x^{2} + 15)}^{\frac{2}{3}})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.13.2

Move $50$ to the left of $x^{3}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x + \frac{50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.14

To write $- 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x - 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x \cdot \frac{3}{3} + \frac{50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.15

Combine $- 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ and $\frac{3}{3}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{- 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x \cdot 3}{3} + \frac{50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.16

Combine the numerators over the common denominator.

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{- 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x \cdot 3 + 50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17

Simplify the numerator.

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Step 3.18.4.17.1

Factor $50 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $- 150 {(5 x^{2} + 15)}^{\frac{2}{3}} x \cdot 3 + 50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}$ .

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Step 3.18.4.17.1.1

Reorder the expression.

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Step 3.18.4.17.1.1.1

Move ${(5 x^{2} + 15)}^{\frac{2}{3}}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{- 150 x \cdot (3 {(5 x^{2} + 15)}^{\frac{2}{3}}) + 50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.1.1.2

Move $x$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{- 150 \cdot (3 x {(5 x^{2} + 15)}^{\frac{2}{3}}) + 50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.1.2

Factor $50 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $- 150 \cdot 3 x {(5 x^{2} + 15)}^{\frac{2}{3}}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 \cdot 3) + 50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.1.3

Factor $50 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $50 x^{3} {(5 x^{2} + 15)}^{\frac{2}{3}}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 \cdot 3) + 50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.1.4

Factor $50 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 \cdot 3) + 50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2})$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 \cdot 3 + x^{2})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.2

Multiply $- 3$ by $3$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 9 + x^{2})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.3

Rewrite $9$ as $3^{2}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3^{2} + x^{2})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.4

Reorder $- 3^{2}$ and $x^{2}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 3^{2})}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.17.5

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 $b = 3$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.18

To write $- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f''' (x) = \frac{50 (- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x \cdot \frac{3}{3} + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.19

Combine $- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x$ and $\frac{3}{3}$ .

$f''' (x) = \frac{50 (\frac{- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x \cdot 3}{3} + \frac{50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.20

Combine the numerators over the common denominator.

$f''' (x) = \frac{50 (\frac{- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x \cdot 3 + 50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.21

Rewrite $\frac{- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x \cdot 3 + 50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)}{3}$ in a factored form.

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Step 3.18.4.21.1

Factor $2 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $- 2 {(5 x^{2} + 15)}^{\frac{5}{3}} x \cdot 3 + 50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)$ .

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Step 3.18.4.21.1.1

Reorder the expression.

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Step 3.18.4.21.1.1.1

Move ${(5 x^{2} + 15)}^{\frac{5}{3}}$ .

$f''' (x) = \frac{50 (\frac{- 2 x \cdot (3 {(5 x^{2} + 15)}^{\frac{5}{3}}) + 50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.21.1.1.2

Move $x$ .

$f''' (x) = \frac{50 (\frac{- 2 \cdot (3 x {(5 x^{2} + 15)}^{\frac{5}{3}}) + 50 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.21.1.1.3

Move ${(5 x^{2} + 15)}^{\frac{2}{3}}$ .

$f''' (x) = \frac{50 (\frac{- 2 \cdot (3 x {(5 x^{2} + 15)}^{\frac{5}{3}}) + 50 x {(5 x^{2} + 15)}^{\frac{2}{3}} (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.21.1.2

Factor $2 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $- 2 \cdot 3 x {(5 x^{2} + 15)}^{\frac{5}{3}}$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 1 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}})) + 50 x {(5 x^{2} + 15)}^{\frac{2}{3}} (x + 3) (x - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.21.1.3

Factor $2 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $50 x {(5 x^{2} + 15)}^{\frac{2}{3}} (x + 3) (x - 3)$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 1 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}})) + 2 {(5 x^{2} + 15)}^{\frac{2}{3}} x ((25 (x + 3)) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.21.1.4

Factor $2 {(5 x^{2} + 15)}^{\frac{2}{3}} x$ out of $2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 1 \cdot 3 {(5 x^{2} + 15)}^{\frac{3}{3}}) + 2 {(5 x^{2} + 15)}^{\frac{2}{3}} x ((25 (x + 3)) (x - 3))$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 1 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}) + (25 (x + 3)) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.21.2

Multiply $- 1$ by $3$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 {(5 x^{2} + 15)}^{\frac{3}{3}} + (25 (x + 3)) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.22

Cancel the common factor of $3$ .

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Step 3.18.4.22.1

Cancel the common factor.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 {(5 x^{2} + 15)}^{\frac{3}{3}} + (25 (x + 3)) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.22.2

Rewrite the expression.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 (5 x^{2} + 15) + (25 (x + 3)) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23

Simplify the numerator.

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Step 3.18.4.23.1

Simplify.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 (5 x^{2} + 15) + 25 (x + 3) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.2

Apply the distributive property.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 3 (5 x^{2}) - 3 \cdot 15 + 25 (x + 3) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.3

Multiply $5$ by $- 3$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 3 \cdot 15 + 25 (x + 3) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.4

Multiply $- 3$ by $15$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 (x + 3) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.5

Apply the distributive property.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + (25 x + 25 \cdot 3) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.6

Multiply $25$ by $3$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + (25 x + 75) (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.7

Expand $(25 x + 75) (x - 3)$ using the FOIL Method.

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Step 3.18.4.23.7.1

Apply the distributive property.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x (x - 3) + 75 (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.7.2

Apply the distributive property.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x \cdot x + 25 x \cdot - 3 + 75 (x - 3))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.7.3

Apply the distributive property.

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x \cdot x + 25 x \cdot - 3 + 75 x + 75 \cdot - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.8

Simplify and combine like terms.

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Step 3.18.4.23.8.1

Simplify each term.

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Step 3.18.4.23.8.1.1

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

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Step 3.18.4.23.8.1.1.1

Move $x$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 (x \cdot x) + 25 x \cdot - 3 + 75 x + 75 \cdot - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.8.1.1.2

Multiply $x$ by $x$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x^{2} + 25 x \cdot - 3 + 75 x + 75 \cdot - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.8.1.2

Multiply $- 3$ by $25$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x^{2} - 75 x + 75 x + 75 \cdot - 3)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.8.1.3

Multiply $75$ by $- 3$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x^{2} - 75 x + 75 x - 225)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.8.2

Add $- 75 x$ and $75 x$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x^{2} + 0 - 225)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.8.3

Add $25 x^{2}$ and $0$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (- 15 x^{2} - 45 + 25 x^{2} - 225)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.9

Add $- 15 x^{2}$ and $25 x^{2}$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (10 x^{2} - 45 - 225)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.10

Subtract $225$ from $- 45$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (10 x^{2} - 270)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.11

Factor $10$ out of $10 x^{2} - 270$ .

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Step 3.18.4.23.11.1

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

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (10 (x^{2}) - 270)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.11.2

Factor $10$ out of $- 270$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (10 x^{2} + 10 \cdot - 27)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.11.3

Factor $10$ out of $10 x^{2} + 10 \cdot - 27$ .

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x (10 (x^{2} - 27))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

$f''' (x) = \frac{50 (\frac{2 {(5 x^{2} + 15)}^{\frac{2}{3}} x \cdot (10 (x^{2} - 27))}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.4.23.12

Multiply $10$ by $2$ .

$f''' (x) = \frac{50 (\frac{20 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27)}{3})}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.5

Combine terms.

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Step 3.18.5.1

Combine $50$ and $\frac{20 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27)}{3}$ .

$f''' (x) = \frac{\frac{50 (20 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27))}{3}}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.5.2

Multiply $20$ by $50$ .

$f''' (x) = \frac{\frac{1000 ({(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27))}{3}}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.5.3

Rewrite $\frac{\frac{1000 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27)}{3}}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$ as a product.

$f''' (x) = \frac{1000 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27)}{3} \cdot \frac{1}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.5.4

Multiply $\frac{1000 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27)}{3}$ by $\frac{1}{9 {(5 x^{2} + 15)}^{\frac{10}{3}}}$ .

$f''' (x) = \frac{1000 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27)}{3 (9 {(5 x^{2} + 15)}^{\frac{10}{3}})}$

Step 3.18.5.5

Multiply $9$ by $3$ .

$f''' (x) = \frac{1000 {(5 x^{2} + 15)}^{\frac{2}{3}} x (x^{2} - 27)}{27 {(5 x^{2} + 15)}^{\frac{10}{3}}}$

Step 3.18.5.6

Move ${(5 x^{2} + 15)}^{\frac{2}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$f''' (x) = \frac{1000 x (x^{2} - 27)}{27 {(5 x^{2} + 15)}^{\frac{10}{3}} {(5 x^{2} + 15)}^{- \frac{2}{3}}}$

Step 3.18.5.7

Multiply ${(5 x^{2} + 15)}^{\frac{10}{3}}$ by ${(5 x^{2} + 15)}^{- \frac{2}{3}}$ by adding the exponents.

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Step 3.18.5.7.1

Move ${(5 x^{2} + 15)}^{- \frac{2}{3}}$ .

$f''' (x) = \frac{1000 x (x^{2} - 27)}{27 ({(5 x^{2} + 15)}^{- \frac{2}{3}} {(5 x^{2} + 15)}^{\frac{10}{3}})}$

Step 3.18.5.7.2

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

$f''' (x) = \frac{1000 x (x^{2} - 27)}{27 {(5 x^{2} + 15)}^{- \frac{2}{3} + \frac{10}{3}}}$

Step 3.18.5.7.3

Combine the numerators over the common denominator.

$f''' (x) = \frac{1000 x (x^{2} - 27)}{27 {(5 x^{2} + 15)}^{\frac{- 2 + 10}{3}}}$

Step 3.18.5.7.4

Add $- 2$ and $10$ .

$f''' (x) = \frac{1000 x (x^{2} - 27)}{27 {(5 x^{2} + 15)}^{\frac{8}{3}}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{1000}{27}$ is constant with respect to $x$ , the derivative of $\frac{1000 x (x^{2} - 27)}{27 {(5 x^{2} + 15)}^{\frac{8}{3}}}$ with respect to $x$ is $\frac{1000}{27} \frac{d}{d x} [\frac{x (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{8}{3}}}]$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{d}{d x} (\frac{x (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{8}{3}}})$

Step 4.2

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} \frac{d}{d x} (x (x^{2} - 27)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{({(5 x^{2} + 15)}^{\frac{8}{3}})}^{2}}$

Step 4.3

Multiply the exponents in ${({(5 x^{2} + 15)}^{\frac{8}{3}})}^{2}$ .

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Step 4.3.1

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} \frac{d}{d x} (x (x^{2} - 27)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{8}{3} \cdot 2}}$

Step 4.3.2

Multiply $\frac{8}{3} \cdot 2$ .

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Step 4.3.2.1

Combine $\frac{8}{3}$ and $2$ .

Step 4.3.2.2

Multiply $8$ by $2$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} \frac{d}{d x} (x (x^{2} - 27)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.4

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 $f (x) = x$ and $g (x) = x^{2} - 27$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (x \frac{d}{d x} (x^{2} - 27) + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.5

Differentiate.

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Step 4.5.1

By the Sum Rule, the derivative of $x^{2} - 27$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 27]$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 27)) + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.5.2

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (x (2 x + \frac{d}{d x} (- 27)) + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.5.3

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27$ with respect to $x$ is $0$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (x (2 x + 0) + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.5.4

Add $2 x$ and $0$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (x (2 x) + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.6

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (2 (x \cdot x) + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.7

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

Step 4.8

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (2 x^{1 + 1} + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.9

Differentiate using the Power Rule.

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Step 4.9.1

Add $1$ and $1$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (2 x^{2} + (x^{2} - 27) \frac{d}{d x} (x)) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.9.2

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (2 x^{2} + (x^{2} - 27) \cdot 1) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.9.3

Simplify by adding terms.

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Step 4.9.3.1

Multiply $x^{2} - 27$ by $1$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (2 x^{2} + x^{2} - 27) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.9.3.2

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) \frac{d}{d x} {(5 x^{2} + 15)}^{\frac{8}{3}}}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.10

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{8}{3}}$ and $g (x) = 5 x^{2} + 15$ .

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Step 4.10.1

To apply the Chain Rule, set $u$ as $5 x^{2} + 15$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{d}{d u} (u^{\frac{8}{3}}) \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.10.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{8}{3}$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8}{3} u^{\frac{8}{3} - 1} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.10.3

Replace all occurrences of $u$ with $5 x^{2} + 15$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8}{3} \cdot {(5 x^{2} + 15)}^{\frac{8}{3} - 1} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.11

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8}{3} \cdot {(5 x^{2} + 15)}^{\frac{8}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.12

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8}{3} \cdot {(5 x^{2} + 15)}^{\frac{8}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.13

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8}{3} \cdot {(5 x^{2} + 15)}^{\frac{8 - 1 \cdot 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.14

Simplify the numerator.

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Step 4.14.1

Multiply $- 1$ by $3$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8}{3} \cdot {(5 x^{2} + 15)}^{\frac{8 - 3}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.14.2

Subtract $3$ from $8$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8}{3} \cdot {(5 x^{2} + 15)}^{\frac{5}{3}} \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.15

Combine fractions.

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Step 4.15.1

Combine $\frac{8}{3}$ and ${(5 x^{2} + 15)}^{\frac{5}{3}}$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - x (x^{2} - 27) (\frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot \frac{d}{d x} (5 x^{2} + 15))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.15.2

Combine $\frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}}}{3}$ and $x$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot (x^{2} - 27) \frac{d}{d x} (5 x^{2} + 15)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.16

By the Sum Rule, the derivative of $5 x^{2} + 15$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [15]$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot ((x^{2} - 27) (\frac{d}{d x} (5 x^{2}) + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.17

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot ((x^{2} - 27) (5 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.18

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot ((x^{2} - 27) (5 (2 x) + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.19

Multiply $2$ by $5$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot ((x^{2} - 27) (10 x + \frac{d}{d x} (15)))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.20

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot ((x^{2} - 27) (10 x + 0))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.21

Combine fractions.

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Step 4.21.1

Add $10 x$ and $0$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot ((x^{2} - 27) (10 x))}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.21.2

Multiply $10$ by $- 1$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - 10 \frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3} \cdot ((x^{2} - 27) x)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.21.3

Combine $- 10$ and $\frac{8 {(5 x^{2} + 15)}^{\frac{5}{3}} x}{3}$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) + \frac{- 10 (8 {(5 x^{2} + 15)}^{\frac{5}{3}} x)}{3} \cdot ((x^{2} - 27) x)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.21.4

Multiply $8$ by $- 10$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) + \frac{- 80 ({(5 x^{2} + 15)}^{\frac{5}{3}} x)}{3} \cdot ((x^{2} - 27) x)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.21.5

Combine $x$ and $\frac{- 80 ({(5 x^{2} + 15)}^{\frac{5}{3}} x)}{3}$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) + \frac{x (- 80 ({(5 x^{2} + 15)}^{\frac{5}{3}} x))}{3} \cdot (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.22

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) + \frac{x \cdot x (- 80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} \cdot (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.23

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

Step 4.24

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

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) + \frac{x^{1 + 1} (- 80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} \cdot (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.25

Simplify the expression.

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Step 4.25.1

Add $1$ and $1$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) + \frac{x^{2} (- 80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} \cdot (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.25.2

Move the negative in front of the fraction.

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{x^{2} \cdot (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} \cdot (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.25.3

Move $80$ to the left of $x^{2}$ .

$f^{4} (x) = \frac{1000}{27} \cdot \frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{80 \cdot (x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} \cdot (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.26

Multiply $\frac{1000}{27}$ by $\frac{{(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} (x^{2} - 27)}{{(5 x^{2} + 15)}^{\frac{16}{3}}}$ .

$f^{4} (x) = \frac{1000 ({(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot (x^{2} - 27))}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27

Simplify.

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Step 4.27.1

Apply the distributive property.

$f^{4} (x) = \frac{1000 ({(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27)) + 1000 (- \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot (x^{2} - 27))}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2

Simplify the numerator.

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Step 4.27.2.1

Factor $1000$ out of $1000 {(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) + 1000 (- \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} (x^{2} - 27))$ .

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Step 4.27.2.1.1

Factor $1000$ out of $1000 {(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27)$ .

$f^{4} (x) = \frac{1000 ({(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27)) + 1000 (- \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot (x^{2} - 27))}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.1.2

Factor $1000$ out of $1000 ({(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27)) + 1000 (- \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} (x^{2} - 27))$ .

$f^{4} (x) = \frac{1000 ({(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2} - 27) - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot (x^{2} - 27))}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.2

Apply the distributive property.

$f^{4} (x) = \frac{1000 ({(5 x^{2} + 15)}^{\frac{8}{3}} (3 x^{2}) + {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot - 27 - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot (x^{2} - 27))}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.3

Rewrite using the commutative property of multiplication.

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} + {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot - 27 - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot (x^{2} - 27))}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.4

Move $- 27$ to the left of ${(5 x^{2} + 15)}^{\frac{8}{3}}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 \cdot {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot (x^{2} - 27))}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.5

Apply the distributive property.

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot x^{2} - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot - 27)}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.6

Multiply $- \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} x^{2}$ .

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Step 4.27.2.6.1

Combine $x^{2}$ and $\frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{2} (80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot - 27)}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.6.2

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

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Step 4.27.2.6.2.1

Move $x^{2}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{2} x^{2} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot - 27)}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.6.2.2

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

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{2 + 2} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot - 27)}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.6.2.3

Add $2$ and $2$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{4} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} - \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot - 27)}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.7

Cancel the common factor of $3$ .

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Step 4.27.2.7.1

Move the leading negative in $- \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3}$ into the numerator.

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{4} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} + \frac{- 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} \cdot - 27)}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.7.2

Factor $3$ out of $- 27$ .

Step 4.27.2.7.3

Cancel the common factor.

Step 4.27.2.7.4

Rewrite the expression.

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{4} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} - 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} \cdot - 9)}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.8

Multiply $- 9$ by $- 80$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{4} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.9

Simplify each term.

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Step 4.27.2.9.1

Simplify the numerator.

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Step 4.27.2.9.1.1

Rewrite.

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{2} (80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.9.1.2

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

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Step 4.27.2.9.1.2.1

Move $x^{2}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{2} x^{2} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.9.1.2.2

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

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{2 + 2} (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.9.1.2.3

Add $2$ and $2$ .

Step 4.27.2.9.1.3

Remove unnecessary parentheses.

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{x^{4} \cdot (80 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.9.2

Move $80$ to the left of $x^{4}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{80 x^{4} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.10

To write $720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

Step 4.27.2.11

Combine $720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}$ and $\frac{3}{3}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} - \frac{80 x^{4} {(5 x^{2} + 15)}^{\frac{5}{3}}}{3} + \frac{720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} \cdot 3}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.12

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} + \frac{- 80 x^{4} {(5 x^{2} + 15)}^{\frac{5}{3}} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} \cdot 3}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.13

Simplify the numerator.

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Step 4.27.2.13.1

Factor $80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $- 80 x^{4} {(5 x^{2} + 15)}^{\frac{5}{3}} + 720 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} \cdot 3$ .

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Step 4.27.2.13.1.1

Reorder the expression.

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Step 4.27.2.13.1.1.1

Move ${(5 x^{2} + 15)}^{\frac{5}{3}}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} + \frac{- 80 x^{4} {(5 x^{2} + 15)}^{\frac{5}{3}} + 720 x^{2} \cdot (3 {(5 x^{2} + 15)}^{\frac{5}{3}})}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.13.1.1.2

Move $x^{2}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} + \frac{- 80 x^{4} {(5 x^{2} + 15)}^{\frac{5}{3}} + 720 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.13.1.2

Factor $80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $- 80 x^{4} {(5 x^{2} + 15)}^{\frac{5}{3}}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} + \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2}) + 720 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}})}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.13.1.3

Factor $80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $720 \cdot 3 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} + \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2}) + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (9 \cdot 3)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.13.1.4

Factor $80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2}) + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (9 \cdot 3)$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} + \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 9 \cdot 3)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.13.2

Multiply $9$ by $3$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} + \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.14

To write $3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{4} (x) = \frac{1000 (3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot \frac{3}{3} + \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27)}{3} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.15

Combine $3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2}$ and $\frac{3}{3}$ .

$f^{4} (x) = \frac{1000 (\frac{3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot 3}{3} + \frac{80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27)}{3} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.16

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{1000 (\frac{3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot 3 + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27)}{3} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.17

To write $- 27 {(5 x^{2} + 15)}^{\frac{8}{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f^{4} (x) = \frac{1000 (\frac{3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot 3 + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27)}{3} - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot \frac{3}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.18

Combine $- 27 {(5 x^{2} + 15)}^{\frac{8}{3}}$ and $\frac{3}{3}$ .

$f^{4} (x) = \frac{1000 (\frac{3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot 3 + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27)}{3} + \frac{- 27 {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot 3}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.19

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{1000 (\frac{3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot 3 + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27) - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot 3}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20

Rewrite $\frac{3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot 3 + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27) - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot 3}{3}$ in a factored form.

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Step 4.27.2.20.1

Factor ${(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $3 {(5 x^{2} + 15)}^{\frac{8}{3}} x^{2} \cdot 3 + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27) - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot 3$ .

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Step 4.27.2.20.1.1

Reorder the expression.

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Step 4.27.2.20.1.1.1

Move ${(5 x^{2} + 15)}^{\frac{8}{3}}$ .

$f^{4} (x) = \frac{1000 (\frac{3 x^{2} \cdot (3 {(5 x^{2} + 15)}^{\frac{8}{3}}) + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27) - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot 3}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.1.1.2

Move $x^{2}$ .

$f^{4} (x) = \frac{1000 (\frac{3 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{8}{3}}) + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27) - 27 {(5 x^{2} + 15)}^{\frac{8}{3}} \cdot 3}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.1.1.3

Move ${(5 x^{2} + 15)}^{\frac{8}{3}}$ .

$f^{4} (x) = \frac{1000 (\frac{3 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{8}{3}}) + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{8}{3}})}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.1.2

Factor ${(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $3 \cdot 3 x^{2} {(5 x^{2} + 15)}^{\frac{8}{3}}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (3 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}})) + 80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{8}{3}})}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.1.3

Factor ${(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $80 x^{2} {(5 x^{2} + 15)}^{\frac{5}{3}} (- x^{2} + 27)$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (3 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}})) + {(5 x^{2} + 15)}^{\frac{5}{3}} (80 x^{2} (- x^{2} + 27)) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{8}{3}})}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.1.4

Factor ${(5 x^{2} + 15)}^{\frac{5}{3}}$ out of $- 27 \cdot 3 {(5 x^{2} + 15)}^{\frac{8}{3}}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (3 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}})) + {(5 x^{2} + 15)}^{\frac{5}{3}} (80 x^{2} (- x^{2} + 27)) + {(5 x^{2} + 15)}^{\frac{5}{3}} (- 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.1.5

Factor ${(5 x^{2} + 15)}^{\frac{5}{3}}$ out of ${(5 x^{2} + 15)}^{\frac{5}{3}} (3 \cdot 3 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}}) + {(5 x^{2} + 15)}^{\frac{5}{3}} (80 x^{2} (- x^{2} + 27))$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (3 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}}) + 80 x^{2} (- x^{2} + 27)) + {(5 x^{2} + 15)}^{\frac{5}{3}} (- 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.1.6

Factor ${(5 x^{2} + 15)}^{\frac{5}{3}}$ out of ${(5 x^{2} + 15)}^{\frac{5}{3}} (3 \cdot 3 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}} + 80 x^{2} (- x^{2} + 27)) + {(5 x^{2} + 15)}^{\frac{5}{3}} (- 27 \cdot 3 {(5 x^{2} + 15)}^{\frac{3}{3}})$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (3 \cdot (3 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}}) + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.2

Multiply $3$ by $3$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 x^{2} {(5 x^{2} + 15)}^{\frac{3}{3}} + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.3

Divide $3$ by $3$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 x^{2} (5 x^{2} + 15) + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.4

Simplify.

Step 4.27.2.20.5

Apply the distributive property.

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 x^{2} (5 x^{2}) + 9 x^{2} \cdot 15 + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.6

Rewrite using the commutative property of multiplication.

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 \cdot (5 x^{2} x^{2}) + 9 x^{2} \cdot 15 + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.7

Multiply $15$ by $9$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 \cdot (5 x^{2} x^{2}) + 135 x^{2} + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.8

Simplify each term.

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Step 4.27.2.20.8.1

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

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Step 4.27.2.20.8.1.1

Move $x^{2}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 \cdot (5 (x^{2} x^{2})) + 135 x^{2} + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.8.1.2

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

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 \cdot (5 x^{2 + 2}) + 135 x^{2} + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.8.1.3

Add $2$ and $2$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (9 \cdot (5 x^{4}) + 135 x^{2} + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.8.2

Multiply $9$ by $5$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} + 80 x^{2} (- x^{2} + 27) - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.9

Apply the distributive property.

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} + 80 x^{2} (- x^{2}) + 80 x^{2} \cdot 27 - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.10

Rewrite using the commutative property of multiplication.

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} + 80 \cdot (- 1 x^{2} x^{2}) + 80 x^{2} \cdot 27 - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.11

Multiply $27$ by $80$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} + 80 \cdot (- 1 x^{2} x^{2}) + 2160 x^{2} - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.12

Simplify each term.

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Step 4.27.2.20.12.1

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

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Step 4.27.2.20.12.1.1

Move $x^{2}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} + 80 \cdot (- 1 (x^{2} x^{2})) + 2160 x^{2} - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.12.1.2

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

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} + 80 \cdot (- 1 x^{2 + 2}) + 2160 x^{2} - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.12.1.3

Add $2$ and $2$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} + 80 \cdot (- 1 x^{4}) + 2160 x^{2} - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.12.2

Multiply $80$ by $- 1$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} - 80 x^{4} + 2160 x^{2} - 27 \cdot (3 {(5 x^{2} + 15)}^{\frac{3}{3}}))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.13

Multiply $- 27$ by $3$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} - 80 x^{4} + 2160 x^{2} - 81 {(5 x^{2} + 15)}^{\frac{3}{3}})}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.14

Divide $3$ by $3$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} - 80 x^{4} + 2160 x^{2} - 81 (5 x^{2} + 15))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.15

Simplify.

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} - 80 x^{4} + 2160 x^{2} - 81 (5 x^{2} + 15))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.16

Apply the distributive property.

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} - 80 x^{4} + 2160 x^{2} - 81 (5 x^{2}) - 81 \cdot 15)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.17

Multiply $5$ by $- 81$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} - 80 x^{4} + 2160 x^{2} - 405 x^{2} - 81 \cdot 15)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.18

Multiply $- 81$ by $15$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (45 x^{4} + 135 x^{2} - 80 x^{4} + 2160 x^{2} - 405 x^{2} - 1215)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.19

Subtract $80 x^{4}$ from $45 x^{4}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- 35 x^{4} + 135 x^{2} + 2160 x^{2} - 405 x^{2} - 1215)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.20

Add $135 x^{2}$ and $2160 x^{2}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- 35 x^{4} + 2295 x^{2} - 405 x^{2} - 1215)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.21

Subtract $405 x^{2}$ from $2295 x^{2}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (- 35 x^{4} + 1890 x^{2} - 1215)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.22

Factor $5$ out of $- 35 x^{4} + 1890 x^{2} - 1215$ .

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Step 4.27.2.20.22.1

Factor $5$ out of $- 35 x^{4}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (5 (- 7 x^{4}) + 1890 x^{2} - 1215)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.22.2

Factor $5$ out of $1890 x^{2}$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (5 (- 7 x^{4}) + 5 (378 x^{2}) - 1215)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.22.3

Factor $5$ out of $- 1215$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (5 (- 7 x^{4}) + 5 (378 x^{2}) + 5 (- 243))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.22.4

Factor $5$ out of $5 (- 7 x^{4}) + 5 (378 x^{2})$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (5 (- 7 x^{4} + 378 x^{2}) + 5 (- 243))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.20.22.5

Factor $5$ out of $5 (- 7 x^{4} + 378 x^{2}) + 5 (- 243)$ .

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} (5 (- 7 x^{4} + 378 x^{2} - 243))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

$f^{4} (x) = \frac{1000 (\frac{{(5 x^{2} + 15)}^{\frac{5}{3}} \cdot (5 (- 7 x^{4} + 378 x^{2} - 243))}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.2.21

Move $5$ to the left of ${(5 x^{2} + 15)}^{\frac{5}{3}}$ .

$f^{4} (x) = \frac{1000 (\frac{5 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243)}{3})}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.3

Combine terms.

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Step 4.27.3.1

Combine $1000$ and $\frac{5 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243)}{3}$ .

$f^{4} (x) = \frac{\frac{1000 (5 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243))}{3}}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.3.2

Multiply $5$ by $1000$ .

$f^{4} (x) = \frac{\frac{5000 ({(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243))}{3}}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.3.3

Rewrite $\frac{\frac{5000 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243)}{3}}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$ as a product.

$f^{4} (x) = \frac{5000 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243)}{3} \cdot \frac{1}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.3.4

Multiply $\frac{5000 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243)}{3}$ by $\frac{1}{27 {(5 x^{2} + 15)}^{\frac{16}{3}}}$ .

$f^{4} (x) = \frac{5000 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243)}{3 (27 {(5 x^{2} + 15)}^{\frac{16}{3}})}$

Step 4.27.3.5

Multiply $27$ by $3$ .

$f^{4} (x) = \frac{5000 {(5 x^{2} + 15)}^{\frac{5}{3}} (- 7 x^{4} + 378 x^{2} - 243)}{81 {(5 x^{2} + 15)}^{\frac{16}{3}}}$

Step 4.27.3.6

Move ${(5 x^{2} + 15)}^{\frac{5}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$f^{4} (x) = \frac{5000 (- 7 x^{4} + 378 x^{2} - 243)}{81 {(5 x^{2} + 15)}^{\frac{16}{3}} {(5 x^{2} + 15)}^{- \frac{5}{3}}}$

Step 4.27.3.7

Multiply ${(5 x^{2} + 15)}^{\frac{16}{3}}$ by ${(5 x^{2} + 15)}^{- \frac{5}{3}}$ by adding the exponents.

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Step 4.27.3.7.1

Move ${(5 x^{2} + 15)}^{- \frac{5}{3}}$ .

$f^{4} (x) = \frac{5000 (- 7 x^{4} + 378 x^{2} - 243)}{81 ({(5 x^{2} + 15)}^{- \frac{5}{3}} {(5 x^{2} + 15)}^{\frac{16}{3}})}$

Step 4.27.3.7.2

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

$f^{4} (x) = \frac{5000 (- 7 x^{4} + 378 x^{2} - 243)}{81 {(5 x^{2} + 15)}^{- \frac{5}{3} + \frac{16}{3}}}$

Step 4.27.3.7.3

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{5000 (- 7 x^{4} + 378 x^{2} - 243)}{81 {(5 x^{2} + 15)}^{\frac{- 5 + 16}{3}}}$

Step 4.27.3.7.4

Add $- 5$ and $16$ .

$f^{4} (x) = \frac{5000 (- 7 x^{4} + 378 x^{2} - 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 4.27.4

Factor $- 1$ out of $- 7 x^{4}$ .

$f^{4} (x) = \frac{5000 (- (7 x^{4}) + 378 x^{2} - 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 4.27.5

Factor $- 1$ out of $378 x^{2}$ .

$f^{4} (x) = \frac{5000 (- (7 x^{4}) - (- 378 x^{2}) - 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 4.27.6

Factor $- 1$ out of $- (7 x^{4}) - (- 378 x^{2})$ .

$f^{4} (x) = \frac{5000 (- (7 x^{4} - 378 x^{2}) - 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 4.27.7

Rewrite $- 243$ as $- 1 (243)$ .

$f^{4} (x) = \frac{5000 (- (7 x^{4} - 378 x^{2}) - 1 \cdot 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 4.27.8

Factor $- 1$ out of $- (7 x^{4} - 378 x^{2}) - 1 (243)$ .

$f^{4} (x) = \frac{5000 (- (7 x^{4} - 378 x^{2} + 243))}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 4.27.9

Rewrite $- (7 x^{4} - 378 x^{2} + 243)$ as $- 1 (7 x^{4} - 378 x^{2} + 243)$ .

$f^{4} (x) = \frac{5000 (- 1 (7 x^{4} - 378 x^{2} + 243))}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 4.27.10

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{5000 (7 x^{4} - 378 x^{2} + 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{5000 (7 x^{4} - 378 x^{2} + 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$ .

$- \frac{5000 (7 x^{4} - 378 x^{2} + 243)}{81 {(5 x^{2} + 15)}^{\frac{11}{3}}}$