Calculus Examples

$y = \frac{x^{3}}{\sqrt{x^{2} - 4}}$

Step 1

Write $y = \frac{x^{3}}{\sqrt{x^{2} - 4}}$ as a function.

$f (x) = \frac{x^{3}}{\sqrt{x^{2} - 4}}$

Step 2

Find the first derivative of the function.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{x^{3}}{{(x^{2} - 4)}^{\frac{1}{2}}}]$

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 $f (x) = x^{3}$ and $g (x) = {(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{({(x^{2} - 4)}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(x^{2} - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1

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

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{(x^{2} - 4)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{(x^{2} - 4)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.2

Rewrite the expression.

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{(x^{2} - 4)}^{1}}$

Step 2.4

Simplify.

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{x^{2} - 4}$

Step 2.5

Differentiate using the Power Rule.

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Step 2.5.1

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

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} (3 x^{2}) - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{x^{2} - 4}$

Step 2.5.2

Move $3$ to the left of ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{3 \cdot {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{x^{2} - 4}$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{2} - 4$ .

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Step 2.6.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.6.2

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.6.3

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.8

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.9

Combine the numerators over the common denominator.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.10.2

Subtract $2$ from $1$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.11.2

Combine $\frac{1}{2}$ and ${(x^{2} - 4)}^{- \frac{1}{2}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{{(x^{2} - 4)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.11.3

Move ${(x^{2} - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 2.11.4

Combine $\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}}$ and $x^{3}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} - 4]}{x^{2} - 4}$

Step 2.12

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])}{x^{2} - 4}$

Step 2.13

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [- 4])}{x^{2} - 4}$

Step 2.14

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x + 0)}{x^{2} - 4}$

Step 2.15

Combine fractions.

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Step 2.15.1

Add $2 x$ and $0$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x)}{x^{2} - 4}$

Step 2.15.2

Multiply $2$ by $- 1$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - 2 \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} x}{x^{2} - 4}$

Step 2.15.3

Combine $- 2$ and $\frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} x}{x^{2} - 4}$

Step 2.15.4

Combine $\frac{- 2 x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}$ and $x$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{3} x}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.16

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 (x^{1} x^{3})}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.17

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{1 + 3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.18

Add $1$ and $3$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{4}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.19

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{2 (- x^{4})}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.20

Cancel the common factors.

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Step 2.20.1

Factor $2$ out of $2 {(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{2 (- x^{4})}{2 ({(x^{2} - 4)}^{\frac{1}{2}})}}{x^{2} - 4}$

Step 2.20.2

Cancel the common factor.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{2 (- x^{4})}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.20.3

Rewrite the expression.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.21

Move the negative in front of the fraction.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.22

Combine $3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2}$ and $- \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}$ using a common denominator.

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Step 2.22.1

Move ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}} - \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.22.2

To write $3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} - 4)}^{\frac{1}{2}}}{{(x^{2} - 4)}^{\frac{1}{2}}}$ .

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{\frac{1}{2}}}{{(x^{2} - 4)}^{\frac{1}{2}}} - \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.22.3

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{\frac{1}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.23

Multiply ${(x^{2} - 4)}^{\frac{1}{2}}$ by ${(x^{2} - 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.23.1

Move ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{\frac{3 x^{2} ({(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{\frac{1}{2}}) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.23.2

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

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2} + \frac{1}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.23.3

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1 + 1}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.23.4

Add $1$ and $1$ .

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{2}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.23.5

Divide $2$ by $2$ .

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{1} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.24

Simplify $3 x^{2} {(x^{2} - 4)}^{1}$ .

$\frac{\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 2.25

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}} \cdot \frac{1}{x^{2} - 4}$

Step 2.26

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}} (x^{2} - 4)}$

Step 2.27

Multiply ${(x^{2} - 4)}^{\frac{1}{2}}$ by $x^{2} - 4$ by adding the exponents.

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Step 2.27.1

Multiply ${(x^{2} - 4)}^{\frac{1}{2}}$ by $x^{2} - 4$ .

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Step 2.27.1.1

Raise $x^{2} - 4$ to the power of $1$ .

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{1}}$

Step 2.27.1.2

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2} + 1}}$

Step 2.27.2

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 2.27.3

Combine the numerators over the common denominator.

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1 + 2}{2}}}$

Step 2.27.4

Add $1$ and $2$ .

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28

Simplify.

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Step 2.28.1

Apply the distributive property.

$\frac{3 x^{2} x^{2} + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.2

Simplify the numerator.

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Step 2.28.2.1

Simplify each term.

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Step 2.28.2.1.1

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

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Step 2.28.2.1.1.1

Move $x^{2}$ .

$\frac{3 (x^{2} x^{2}) + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.2.1.1.2

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

$\frac{3 x^{2 + 2} + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.2.1.1.3

Add $2$ and $2$ .

$\frac{3 x^{4} + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.2.1.2

Multiply $- 4$ by $3$ .

$\frac{3 x^{4} - 12 x^{2} - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.2.2

Subtract $x^{4}$ from $3 x^{4}$ .

$\frac{2 x^{4} - 12 x^{2}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.3

Factor $2 x^{2}$ out of $2 x^{4} - 12 x^{2}$ .

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Step 2.28.3.1

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

$\frac{2 x^{2} (x^{2}) - 12 x^{2}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.3.2

Factor $2 x^{2}$ out of $- 12 x^{2}$ .

$\frac{2 x^{2} (x^{2}) + 2 x^{2} (- 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 2.28.3.3

Factor $2 x^{2}$ out of $2 x^{2} (x^{2}) + 2 x^{2} (- 6)$ .

$\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}]$ .

$f'' (x) = 2 \frac{d}{d x} (\frac{x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}})$

Step 3.2

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} \frac{d}{d x} (x^{2} (x^{2} - 6)) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{({(x^{2} - 4)}^{\frac{3}{2}})}^{2}})$

Step 3.3

Multiply the exponents in ${({(x^{2} - 4)}^{\frac{3}{2}})}^{2}$ .

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Step 3.3.1

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} \frac{d}{d x} (x^{2} (x^{2} - 6)) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{\frac{3}{2} \cdot 2}})$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} \frac{d}{d x} (x^{2} (x^{2} - 6)) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{\frac{3}{2} \cdot 2}})$

Step 3.3.2.2

Rewrite the expression.

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} \frac{d}{d x} (x^{2} (x^{2} - 6)) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

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 $f (x) = x^{2}$ and $g (x) = x^{2} - 6$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x^{2} \frac{d}{d x} (x^{2} - 6) + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.5

Differentiate.

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Step 3.5.1

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 6)) + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.5.2

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x^{2} (2 x + \frac{d}{d x} (- 6)) + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.5.3

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x^{2} (2 x + 0) + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.5.4

Add $2 x$ and $0$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x^{2} (2 x) + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.6

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

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Step 3.6.1

Move $x$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x \cdot x^{2} \cdot 2 + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.6.2

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

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Step 3.6.2.1

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x \cdot x^{2} \cdot 2 + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.6.2.2

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x^{1 + 2} \cdot 2 + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.6.3

Add $1$ and $2$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (x^{3} \cdot 2 + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.7

Differentiate using the Power Rule.

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Step 3.7.1

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 \cdot x^{3} + (x^{2} - 6) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.7.2

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + (x^{2} - 6) (2 x)) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.7.3

Move $2$ to the left of $x^{2} - 6$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 \cdot ((x^{2} - 6) x)) - x^{2} (x^{2} - 6) \frac{d}{d x} {(x^{2} - 4)}^{\frac{3}{2}}}{{(x^{2} - 4)}^{3}})$

Step 3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = x^{2} - 4$ .

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Step 3.8.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{d}{d u} (u^{\frac{3}{2}}) \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.8.2

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.8.3

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3}{2} \cdot {(x^{2} - 4)}^{\frac{3}{2} - 1} \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3}{2} \cdot {(x^{2} - 4)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.10

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3}{2} \cdot {(x^{2} - 4)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.11

Combine the numerators over the common denominator.

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3}{2} \cdot {(x^{2} - 4)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.12

Simplify the numerator.

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Step 3.12.1

Multiply $- 1$ by $2$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3}{2} \cdot {(x^{2} - 4)}^{\frac{3 - 2}{2}} \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.12.2

Subtract $2$ from $3$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3}{2} \cdot {(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.13

Combine fractions.

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Step 3.13.1

Combine $\frac{3}{2}$ and ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - x^{2} (x^{2} - 6) (\frac{3 {(x^{2} - 4)}^{\frac{1}{2}}}{2} \cdot \frac{d}{d x} (x^{2} - 4))}{{(x^{2} - 4)}^{3}})$

Step 3.13.2

Combine $\frac{3 {(x^{2} - 4)}^{\frac{1}{2}}}{2}$ and $x^{2}$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - \frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2}}{2} \cdot (x^{2} - 6) \frac{d}{d x} (x^{2} - 4)}{{(x^{2} - 4)}^{3}})$

Step 3.14

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - \frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2}}{2} \cdot ((x^{2} - 6) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 4)))}{{(x^{2} - 4)}^{3}})$

Step 3.15

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

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - \frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2}}{2} \cdot ((x^{2} - 6) (2 x + \frac{d}{d x} (- 4)))}{{(x^{2} - 4)}^{3}})$

Step 3.16

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) - \frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2}}{2} \cdot ((x^{2} - 6) (2 x + 0))}{{(x^{2} - 4)}^{3}})$

Step 3.17

Combine fractions.

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Step 3.17.1

Add $2 x$ and $0$ .

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

Step 3.17.2

Multiply $2$ by $- 1$ .

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

Step 3.17.3

Combine $- 2$ and $\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2}}{2}$ .

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

Step 3.17.4

Multiply $3$ by $- 2$ .

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

Step 3.17.5

Combine $x$ and $\frac{- 6 ({(x^{2} - 4)}^{\frac{1}{2}} x^{2})}{2}$ .

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

Step 3.18

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

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

Step 3.19

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

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

Step 3.20

Add $2$ and $1$ .

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

Step 3.21

Factor $2$ out of $x^{3} (- 6 {(x^{2} - 4)}^{\frac{1}{2}})$ .

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

Step 3.22

Cancel the common factors.

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Step 3.22.1

Factor $2$ out of $2$ .

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

Step 3.22.2

Cancel the common factor.

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

Step 3.22.3

Rewrite the expression.

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

Step 3.22.4

Divide $x^{3} (- 3 {(x^{2} - 4)}^{\frac{1}{2}})$ by $1$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) + x^{3} (- 3 {(x^{2} - 4)}^{\frac{1}{2}}) (x^{2} - 6)}{{(x^{2} - 4)}^{3}})$

Step 3.23

Factor ${(x^{2} - 4)}^{\frac{1}{2}}$ out of ${(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 (x^{2} - 6) x) + x^{3} (- 3 {(x^{2} - 4)}^{\frac{1}{2}}) (x^{2} - 6)$ .

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Step 3.23.1

Reorder the expression.

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Step 3.23.1.1

Move $x^{2} - 6$ .

$f'' (x) = 2 (\frac{{(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 x (x^{2} - 6)) + x^{3} (- 3 {(x^{2} - 4)}^{\frac{1}{2}}) (x^{2} - 6)}{{(x^{2} - 4)}^{3}})$

Step 3.23.1.2

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

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

Step 3.23.2

Factor ${(x^{2} - 4)}^{\frac{1}{2}}$ out of ${(x^{2} - 4)}^{\frac{3}{2}} (2 x^{3} + 2 x (x^{2} - 6))$ .

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

Step 3.23.3

Factor ${(x^{2} - 4)}^{\frac{1}{2}}$ out of $- 3 \cdot x^{3} {(x^{2} - 4)}^{\frac{1}{2}} (x^{2} - 6)$ .

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

Step 3.23.4

Factor ${(x^{2} - 4)}^{\frac{1}{2}}$ out of ${(x^{2} - 4)}^{\frac{1}{2}} ({(x^{2} - 4)}^{\frac{2}{2}} (2 x^{3} + 2 x (x^{2} - 6))) + {(x^{2} - 4)}^{\frac{1}{2}} (- 3 \cdot x^{3} (x^{2} - 6))$ .

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

Step 3.24

Cancel the common factor of $2$ .

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Step 3.24.1

Cancel the common factor.

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

Step 3.24.2

Rewrite the expression.

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

Step 3.25

Simplify.

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

Step 3.26

Move ${(x^{2} - 4)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$f'' (x) = 2 (\frac{(x^{2} - 4) (2 x^{3} + 2 x (x^{2} - 6)) - 3 x^{3} (x^{2} - 6)}{{(x^{2} - 4)}^{3} {(x^{2} - 4)}^{- \frac{1}{2}}})$

Step 3.27

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

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Step 3.27.1

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

$f'' (x) = 2 (\frac{(x^{2} - 4) (2 x^{3} + 2 x (x^{2} - 6)) - 3 x^{3} (x^{2} - 6)}{{(x^{2} - 4)}^{3 - \frac{1}{2}}})$

Step 3.27.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.27.3

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

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

Step 3.27.4

Combine the numerators over the common denominator.

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

Step 3.27.5

Simplify the numerator.

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Step 3.27.5.1

Multiply $3$ by $2$ .

$f'' (x) = 2 (\frac{(x^{2} - 4) (2 x^{3} + 2 x (x^{2} - 6)) - 3 x^{3} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{6 - 1}{2}}})$

Step 3.27.5.2

Subtract $1$ from $6$ .

$f'' (x) = 2 (\frac{(x^{2} - 4) (2 x^{3} + 2 x (x^{2} - 6)) - 3 x^{3} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}})$

Step 3.28

Combine $2$ and $\frac{(x^{2} - 4) (2 x^{3} + 2 x (x^{2} - 6)) - 3 x^{3} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$ .

$f'' (x) = \frac{2 ((x^{2} - 4) (2 x^{3} + 2 x (x^{2} - 6)) - 3 x^{3} (x^{2} - 6))}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29

Simplify.

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Step 3.29.1

Apply the distributive property.

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

Step 3.29.2

Apply the distributive property.

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

Step 3.29.3

Apply the distributive property.

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

Step 3.29.4

Simplify the numerator.

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Step 3.29.4.1

Simplify each term.

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Step 3.29.4.1.1

Simplify each term.

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Step 3.29.4.1.1.1

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

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Step 3.29.4.1.1.1.1

Move $x^{2}$ .

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

Step 3.29.4.1.1.1.2

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

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Step 3.29.4.1.1.1.2.1

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

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

Step 3.29.4.1.1.1.2.2

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

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

Step 3.29.4.1.1.1.3

Add $2$ and $1$ .

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

Step 3.29.4.1.1.2

Multiply $- 6$ by $2$ .

$f'' (x) = \frac{2 ((x^{2} - 4) (2 x^{3} + 2 x^{3} - 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.2

Add $2 x^{3}$ and $2 x^{3}$ .

$f'' (x) = \frac{2 ((x^{2} - 4) (4 x^{3} - 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.3

Expand $(x^{2} - 4) (4 x^{3} - 12 x)$ using the FOIL Method.

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Step 3.29.4.1.3.1

Apply the distributive property.

$f'' (x) = \frac{2 (x^{2} (4 x^{3} - 12 x) - 4 (4 x^{3} - 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.3.2

Apply the distributive property.

$f'' (x) = \frac{2 (x^{2} (4 x^{3}) + x^{2} (- 12 x) - 4 (4 x^{3} - 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.3.3

Apply the distributive property.

$f'' (x) = \frac{2 (x^{2} (4 x^{3}) + x^{2} (- 12 x) - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4

Simplify and combine like terms.

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Step 3.29.4.1.4.1

Simplify each term.

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Step 3.29.4.1.4.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 (4 x^{2} x^{3} + x^{2} (- 12 x) - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.2

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

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Step 3.29.4.1.4.1.2.1

Move $x^{3}$ .

$f'' (x) = \frac{2 (4 (x^{3} x^{2}) + x^{2} (- 12 x) - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.2.2

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

$f'' (x) = \frac{2 (4 x^{3 + 2} + x^{2} (- 12 x) - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.2.3

Add $3$ and $2$ .

$f'' (x) = \frac{2 (4 x^{5} + x^{2} (- 12 x) - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 (4 x^{5} - 12 x^{2} x - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.4

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

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Step 3.29.4.1.4.1.4.1

Move $x$ .

$f'' (x) = \frac{2 (4 x^{5} - 12 (x \cdot x^{2}) - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.4.2

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

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Step 3.29.4.1.4.1.4.2.1

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

$f'' (x) = \frac{2 (4 x^{5} - 12 (x \cdot x^{2}) - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.4.2.2

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

$f'' (x) = \frac{2 (4 x^{5} - 12 x^{1 + 2} - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.4.3

Add $1$ and $2$ .

$f'' (x) = \frac{2 (4 x^{5} - 12 x^{3} - 4 (4 x^{3}) - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.5

Multiply $4$ by $- 4$ .

$f'' (x) = \frac{2 (4 x^{5} - 12 x^{3} - 16 x^{3} - 4 (- 12 x)) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.1.6

Multiply $- 12$ by $- 4$ .

$f'' (x) = \frac{2 (4 x^{5} - 12 x^{3} - 16 x^{3} + 48 x) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.4.2

Subtract $16 x^{3}$ from $- 12 x^{3}$ .

$f'' (x) = \frac{2 (4 x^{5} - 28 x^{3} + 48 x) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.5

Apply the distributive property.

$f'' (x) = \frac{2 (4 x^{5}) + 2 (- 28 x^{3}) + 2 (48 x) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.6

Simplify.

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Step 3.29.4.1.6.1

Multiply $4$ by $2$ .

$f'' (x) = \frac{8 x^{5} + 2 (- 28 x^{3}) + 2 (48 x) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.6.2

Multiply $- 28$ by $2$ .

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 2 (48 x) + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.6.3

Multiply $48$ by $2$ .

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 96 x + 2 (- 3 x^{3} x^{2}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.7

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

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Step 3.29.4.1.7.1

Move $x^{2}$ .

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 96 x + 2 (- 3 (x^{2} x^{3})) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.7.2

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

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 96 x + 2 (- 3 x^{2 + 3}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.7.3

Add $2$ and $3$ .

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 96 x + 2 (- 3 x^{5}) + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.8

Multiply $- 3$ by $2$ .

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 96 x - 6 x^{5} + 2 (- 3 x^{3} \cdot - 6)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.9

Multiply $- 6$ by $- 3$ .

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 96 x - 6 x^{5} + 2 (18 x^{3})}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.1.10

Multiply $18$ by $2$ .

$f'' (x) = \frac{8 x^{5} - 56 x^{3} + 96 x - 6 x^{5} + 36 x^{3}}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.2

Subtract $6 x^{5}$ from $8 x^{5}$ .

$f'' (x) = \frac{2 x^{5} - 56 x^{3} + 96 x + 36 x^{3}}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.4.3

Add $- 56 x^{3}$ and $36 x^{3}$ .

$f'' (x) = \frac{2 x^{5} - 20 x^{3} + 96 x}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.5

Factor $2 x$ out of $2 x^{5} - 20 x^{3} + 96 x$ .

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Step 3.29.5.1

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

$f'' (x) = \frac{2 x (x^{4}) - 20 x^{3} + 96 x}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.5.2

Factor $2 x$ out of $- 20 x^{3}$ .

$f'' (x) = \frac{2 x (x^{4}) + 2 x (- 10 x^{2}) + 96 x}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.5.3

Factor $2 x$ out of $96 x$ .

$f'' (x) = \frac{2 x (x^{4}) + 2 x (- 10 x^{2}) + 2 x (48)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.5.4

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

$f'' (x) = \frac{2 x (x^{4} - 10 x^{2}) + 2 x (48)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 3.29.5.5

Factor $2 x$ out of $2 x (x^{4} - 10 x^{2}) + 2 x (48)$ .

$f'' (x) = \frac{2 x (x^{4} - 10 x^{2} + 48)}{{(x^{2} - 4)}^{\frac{5}{2}}}$

Step 4

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

$\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{x^{3}}{{(x^{2} - 4)}^{\frac{1}{2}}}]$

Step 5.1.2

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{({(x^{2} - 4)}^{\frac{1}{2}})}^{2}}$

Step 5.1.3

Multiply the exponents in ${({(x^{2} - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 5.1.3.1

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

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{(x^{2} - 4)}^{\frac{1}{2} \cdot 2}}$

Step 5.1.3.2

Cancel the common factor of $2$ .

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Step 5.1.3.2.1

Cancel the common factor.

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{(x^{2} - 4)}^{\frac{1}{2} \cdot 2}}$

Step 5.1.3.2.2

Rewrite the expression.

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{{(x^{2} - 4)}^{1}}$

Step 5.1.4

Simplify.

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{x^{2} - 4}$

Step 5.1.5

Differentiate using the Power Rule.

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Step 5.1.5.1

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

$\frac{{(x^{2} - 4)}^{\frac{1}{2}} (3 x^{2}) - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{x^{2} - 4}$

Step 5.1.5.2

Move $3$ to the left of ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{3 \cdot {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} \frac{d}{d x} [{(x^{2} - 4)}^{\frac{1}{2}}]}{x^{2} - 4}$

Step 5.1.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{2} - 4$ .

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Step 5.1.6.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.6.2

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.6.3

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.8

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.9

Combine the numerators over the common denominator.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.10

Simplify the numerator.

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Step 5.1.10.1

Multiply $- 1$ by $2$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.10.2

Subtract $2$ from $1$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.11

Combine fractions.

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Step 5.1.11.1

Move the negative in front of the fraction.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2} {(x^{2} - 4)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.11.2

Combine $\frac{1}{2}$ and ${(x^{2} - 4)}^{- \frac{1}{2}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{{(x^{2} - 4)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.11.3

Move ${(x^{2} - 4)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - x^{3} (\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} - 4])}{x^{2} - 4}$

Step 5.1.11.4

Combine $\frac{1}{2 {(x^{2} - 4)}^{\frac{1}{2}}}$ and $x^{3}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} - 4]}{x^{2} - 4}$

Step 5.1.12

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])}{x^{2} - 4}$

Step 5.1.13

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [- 4])}{x^{2} - 4}$

Step 5.1.14

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x + 0)}{x^{2} - 4}$

Step 5.1.15

Combine fractions.

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Step 5.1.15.1

Add $2 x$ and $0$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} (2 x)}{x^{2} - 4}$

Step 5.1.15.2

Multiply $2$ by $- 1$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - 2 \frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} x}{x^{2} - 4}$

Step 5.1.15.3

Combine $- 2$ and $\frac{x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}} x}{x^{2} - 4}$

Step 5.1.15.4

Combine $\frac{- 2 x^{3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}$ and $x$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{3} x}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.16

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 (x^{1} x^{3})}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.17

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{1 + 3}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.18

Add $1$ and $3$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- 2 x^{4}}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.19

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

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{2 (- x^{4})}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.20

Cancel the common factors.

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Step 5.1.20.1

Factor $2$ out of $2 {(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{2 (- x^{4})}{2 ({(x^{2} - 4)}^{\frac{1}{2}})}}{x^{2} - 4}$

Step 5.1.20.2

Cancel the common factor.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{2 (- x^{4})}{2 {(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.20.3

Rewrite the expression.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} + \frac{- x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.21

Move the negative in front of the fraction.

$\frac{3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2} - \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.22

Combine $3 {(x^{2} - 4)}^{\frac{1}{2}} x^{2}$ and $- \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}$ using a common denominator.

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Step 5.1.22.1

Move ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}} - \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.22.2

To write $3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} - 4)}^{\frac{1}{2}}}{{(x^{2} - 4)}^{\frac{1}{2}}}$ .

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{\frac{1}{2}}}{{(x^{2} - 4)}^{\frac{1}{2}}} - \frac{x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.22.3

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{\frac{1}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.23

Multiply ${(x^{2} - 4)}^{\frac{1}{2}}$ by ${(x^{2} - 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 5.1.23.1

Move ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{\frac{3 x^{2} ({(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{\frac{1}{2}}) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.23.2

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

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1}{2} + \frac{1}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.23.3

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{1 + 1}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.23.4

Add $1$ and $1$ .

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{\frac{2}{2}} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.23.5

Divide $2$ by $2$ .

$\frac{\frac{3 x^{2} {(x^{2} - 4)}^{1} - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.24

Simplify $3 x^{2} {(x^{2} - 4)}^{1}$ .

$\frac{\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}}}{x^{2} - 4}$

Step 5.1.25

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}}} \cdot \frac{1}{x^{2} - 4}$

Step 5.1.26

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}} (x^{2} - 4)}$

Step 5.1.27

Multiply ${(x^{2} - 4)}^{\frac{1}{2}}$ by $x^{2} - 4$ by adding the exponents.

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Step 5.1.27.1

Multiply ${(x^{2} - 4)}^{\frac{1}{2}}$ by $x^{2} - 4$ .

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Step 5.1.27.1.1

Raise $x^{2} - 4$ to the power of $1$ .

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2}} {(x^{2} - 4)}^{1}}$

Step 5.1.27.1.2

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2} + 1}}$

Step 5.1.27.2

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

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 5.1.27.3

Combine the numerators over the common denominator.

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{1 + 2}{2}}}$

Step 5.1.27.4

Add $1$ and $2$ .

$\frac{3 x^{2} (x^{2} - 4) - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28

Simplify.

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Step 5.1.28.1

Apply the distributive property.

$\frac{3 x^{2} x^{2} + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.2

Simplify the numerator.

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Step 5.1.28.2.1

Simplify each term.

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Step 5.1.28.2.1.1

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

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Step 5.1.28.2.1.1.1

Move $x^{2}$ .

$\frac{3 (x^{2} x^{2}) + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.2.1.1.2

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

$\frac{3 x^{2 + 2} + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.2.1.1.3

Add $2$ and $2$ .

$\frac{3 x^{4} + 3 x^{2} \cdot - 4 - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.2.1.2

Multiply $- 4$ by $3$ .

$\frac{3 x^{4} - 12 x^{2} - x^{4}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.2.2

Subtract $x^{4}$ from $3 x^{4}$ .

$\frac{2 x^{4} - 12 x^{2}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.3

Factor $2 x^{2}$ out of $2 x^{4} - 12 x^{2}$ .

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Step 5.1.28.3.1

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

$\frac{2 x^{2} (x^{2}) - 12 x^{2}}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.3.2

Factor $2 x^{2}$ out of $- 12 x^{2}$ .

$\frac{2 x^{2} (x^{2}) + 2 x^{2} (- 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.1.28.3.3

Factor $2 x^{2}$ out of $2 x^{2} (x^{2}) + 2 x^{2} (- 6)$ .

$f' (x) = \frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}$ .

$\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}} = 0$

Step 6.2

Set the numerator equal to zero.

$2 x^{2} (x^{2} - 6) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$x^{2} - 6 = 0$

Step 6.3.2

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 6.3.2.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 6.3.2.2

Solve $x^{2} = 0$ for $x$ .

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Step 6.3.2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 6.3.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.3.2.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.3.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 6.3.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.3.3

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 6.3.3.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 6.3.3.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 6.3.3.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 6.3.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{6}$

Step 6.3.3.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.3.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{6}$

Step 6.3.3.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{6}$

Step 6.3.3.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{6}, - \sqrt{6}$

Step 6.3.4

The final solution is all the values that make $2 x^{2} (x^{2} - 6) = 0$ true.

$x = 0, \sqrt{6}, - \sqrt{6}$

Step 6.4

Exclude the solutions that do not make $\frac{2 x^{2} (x^{2} - 6)}{{(x^{2} - 4)}^{\frac{3}{2}}} = 0$ true.

$x = \sqrt{6}, - \sqrt{6}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{2 x^{2} (x^{2} - 6)}{\sqrt{{(x^{2} - 4)}^{3}}}$

Step 7.2

Set the denominator in $\frac{2 x^{2} (x^{2} - 6)}{\sqrt{{(x^{2} - 4)}^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{{(x^{2} - 4)}^{3}} = 0$

Step 7.3

Solve for $x$ .

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Step 7.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{{(x^{2} - 4)}^{3}}}^{2} = 0^{2}$

Step 7.3.2

Simplify each side of the equation.

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Step 7.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(x^{2} - 4)}^{3}}$ as ${(x^{2} - 4)}^{\frac{3}{2}}$ .

${({(x^{2} - 4)}^{\frac{3}{2}})}^{2} = 0^{2}$

Step 7.3.2.2

Simplify the left side.

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Step 7.3.2.2.1

Multiply the exponents in ${({(x^{2} - 4)}^{\frac{3}{2}})}^{2}$ .

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Step 7.3.2.2.1.1

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

${(x^{2} - 4)}^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 7.3.2.2.1.2

Cancel the common factor of $2$ .

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Step 7.3.2.2.1.2.1

Cancel the common factor.

${(x^{2} - 4)}^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 7.3.2.2.1.2.2

Rewrite the expression.

${(x^{2} - 4)}^{3} = 0^{2}$

Step 7.3.2.3

Simplify the right side.

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Step 7.3.2.3.1

Raising $0$ to any positive power yields $0$ .

${(x^{2} - 4)}^{3} = 0$

Step 7.3.3

Solve for $x$ .

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Step 7.3.3.1

Factor the left side of the equation.

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Step 7.3.3.1.1

Rewrite $4$ as $2^{2}$ .

${(x^{2} - 2^{2})}^{3} = 0$

Step 7.3.3.1.2

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

${((x + 2) (x - 2))}^{3} = 0$

Step 7.3.3.1.3

Apply the product rule to $(x + 2) (x - 2)$ .

${(x + 2)}^{3} {(x - 2)}^{3} = 0$

Step 7.3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 2)}^{3} = 0$

${(x - 2)}^{3} = 0$

Step 7.3.3.3

Set ${(x + 2)}^{3}$ equal to $0$ and solve for $x$ .

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Step 7.3.3.3.1

Set ${(x + 2)}^{3}$ equal to $0$ .

${(x + 2)}^{3} = 0$

Step 7.3.3.3.2

Solve ${(x + 2)}^{3} = 0$ for $x$ .

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Step 7.3.3.3.2.1

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 7.3.3.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 7.3.3.4

Set ${(x - 2)}^{3}$ equal to $0$ and solve for $x$ .

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Step 7.3.3.4.1

Set ${(x - 2)}^{3}$ equal to $0$ .

${(x - 2)}^{3} = 0$

Step 7.3.3.4.2

Solve ${(x - 2)}^{3} = 0$ for $x$ .

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Step 7.3.3.4.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 7.3.3.4.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7.3.3.5

The final solution is all the values that make ${(x + 2)}^{3} {(x - 2)}^{3} = 0$ true.

$x = - 2, 2$

Step 7.4

Set the radicand in $\sqrt{{(x^{2} - 4)}^{3}}$ less than $0$ to find where the expression is undefined.

${(x^{2} - 4)}^{3} < 0$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{{(x^{2} - 4)}^{3}} < \sqrt[3]{0}$

Step 7.5.2

Simplify the equation.

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Step 7.5.2.1

Simplify the left side.

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Step 7.5.2.1.1

Pull terms out from under the radical.

$x^{2} - 4 < \sqrt[3]{0}$

Step 7.5.2.2

Simplify the right side.

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Step 7.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 7.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x^{2} - 4 < \sqrt[3]{0^{3}}$

Step 7.5.2.2.1.2

Pull terms out from under the radical.

$x^{2} - 4 < 0$

Step 7.5.3

Add $4$ to both sides of the inequality.

$x^{2} < 4$

Step 7.5.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} < \sqrt{4}$

Step 7.5.5

Simplify the equation.

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Step 7.5.5.1

Simplify the left side.

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Step 7.5.5.1.1

Pull terms out from under the radical.

$| x | < \sqrt{4}$

Step 7.5.5.2

Simplify the right side.

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Step 7.5.5.2.1

Simplify $\sqrt{4}$ .

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Step 7.5.5.2.1.1

Rewrite $4$ as $2^{2}$ .

$| x | < \sqrt{2^{2}}$

Step 7.5.5.2.1.2

Pull terms out from under the radical.

$| x | < | 2 |$

Step 7.5.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | < 2$

Step 7.5.6

Write $| x | < 2$ as a piecewise.

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Step 7.5.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 7.5.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x < 2$

Step 7.5.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 7.5.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x < 2$

Step 7.5.6.5

Write as a piecewise.

${\begin{cases} x < 2 & x \geq 0 \\ - x < 2 & x < 0 \end{cases}$

Step 7.5.7

Find the intersection of $x < 2$ and $x \geq 0$ .

$0 \leq x < 2$

Step 7.5.8

Solve $- x < 2$ when $x < 0$ .

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Step 7.5.8.1

Divide each term in $- x < 2$ by $- 1$ and simplify.

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Step 7.5.8.1.1

Divide each term in $- x < 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{2}{- 1}$

Step 7.5.8.1.2

Simplify the left side.

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Step 7.5.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{2}{- 1}$

Step 7.5.8.1.2.2

Divide $x$ by $1$ .

$x > \frac{2}{- 1}$

Step 7.5.8.1.3

Simplify the right side.

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Step 7.5.8.1.3.1

Divide $2$ by $- 1$ .

$x > - 2$

Step 7.5.8.2

Find the intersection of $x > - 2$ and $x < 0$ .

$- 2 < x < 0$

Step 7.5.9

Find the union of the solutions.

$- 2 < x < 2$

Step 7.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$- 2 \leq x \leq 2$

$[- 2, 2]$

$- 2 \leq x \leq 2$

$[- 2, 2]$

Step 8

Critical points to evaluate.

$x = \sqrt{6}, - \sqrt{6}$

Step 9

Evaluate the second derivative at $x = \sqrt{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (\sqrt{6}) ({(\sqrt{6})}^{4} - 10 {(\sqrt{6})}^{2} + 48)}{{({(\sqrt{6})}^{2} - 4)}^{\frac{5}{2}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

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Step 10.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{2 \sqrt{6} ({(6^{\frac{1}{2}})}^{4} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.1.2

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

$\frac{2 \sqrt{6} (6^{\frac{1}{2} \cdot 4} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.1.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{2 \sqrt{6} (6^{\frac{4}{2}} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.1.4

Cancel the common factor of $4$ and $2$ .

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Step 10.1.1.4.1

Factor $2$ out of $4$ .

$\frac{2 \sqrt{6} (6^{\frac{2 \cdot 2}{2}} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.1.4.2

Cancel the common factors.

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Step 10.1.1.4.2.1

Factor $2$ out of $2$ .

$\frac{2 \sqrt{6} (6^{\frac{2 \cdot 2}{2 (1)}} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.1.4.2.2

Cancel the common factor.

$\frac{2 \sqrt{6} (6^{\frac{2 \cdot 2}{2 \cdot 1}} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.1.4.2.3

Rewrite the expression.

$\frac{2 \sqrt{6} (6^{\frac{2}{1}} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.1.4.2.4

Divide $2$ by $1$ .

$\frac{2 \sqrt{6} (6^{2} - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.2

Raise $6$ to the power of $2$ .

$\frac{2 \sqrt{6} (36 - 10 {\sqrt{6}}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 10.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{2 \sqrt{6} (36 - 10 {(6^{\frac{1}{2}})}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.3.2

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

$\frac{2 \sqrt{6} (36 - 10 \cdot 6^{\frac{1}{2} \cdot 2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{6} (36 - 10 \cdot 6^{\frac{2}{2}} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.3.4

Cancel the common factor of $2$ .

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Step 10.1.3.4.1

Cancel the common factor.

$\frac{2 \sqrt{6} (36 - 10 \cdot 6^{\frac{2}{2}} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.3.4.2

Rewrite the expression.

$\frac{2 \sqrt{6} (36 - 10 \cdot 6^{1} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.3.5

Evaluate the exponent.

$\frac{2 \sqrt{6} (36 - 10 \cdot 6 + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.4

Multiply $- 10$ by $6$ .

$\frac{2 \sqrt{6} (36 - 60 + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.5

Subtract $60$ from $36$ .

$\frac{2 \sqrt{6} (- 24 + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.6

Add $- 24$ and $48$ .

$\frac{2 \sqrt{6} \cdot 24}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.1.7

Multiply $24$ by $2$ .

$\frac{48 \sqrt{6}}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 10.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{48 \sqrt{6}}{{({(6^{\frac{1}{2}})}^{2} - 4)}^{\frac{5}{2}}}$

Step 10.2.1.2

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

$\frac{48 \sqrt{6}}{{(6^{\frac{1}{2} \cdot 2} - 4)}^{\frac{5}{2}}}$

Step 10.2.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{48 \sqrt{6}}{{(6^{\frac{2}{2}} - 4)}^{\frac{5}{2}}}$

Step 10.2.1.4

Cancel the common factor of $2$ .

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Step 10.2.1.4.1

Cancel the common factor.

$\frac{48 \sqrt{6}}{{(6^{\frac{2}{2}} - 4)}^{\frac{5}{2}}}$

Step 10.2.1.4.2

Rewrite the expression.

$\frac{48 \sqrt{6}}{{(6^{1} - 4)}^{\frac{5}{2}}}$

Step 10.2.1.5

Evaluate the exponent.

$\frac{48 \sqrt{6}}{{(6 - 4)}^{\frac{5}{2}}}$

Step 10.2.2

Subtract $4$ from $6$ .

$\frac{48 \sqrt{6}}{2^{\frac{5}{2}}}$

Step 11

$x = \sqrt{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \sqrt{6}$ is a local minimum

Step 12

Find the y-value when $x = \sqrt{6}$ .

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Step 12.1

Replace the variable $x$ with $\sqrt{6}$ in the expression.

$f (\sqrt{6}) = \frac{{(\sqrt{6})}^{3}}{\sqrt{{(\sqrt{6})}^{2} - 4}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify the numerator.

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Step 12.2.1.1

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$f (\sqrt{6}) = \frac{\sqrt{6^{3}}}{\sqrt{{\sqrt{6}}^{2} - 4}}$

Step 12.2.1.2

Raise $6$ to the power of $3$ .

$f (\sqrt{6}) = \frac{\sqrt{216}}{\sqrt{{\sqrt{6}}^{2} - 4}}$

Step 12.2.1.3

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 12.2.1.3.1

Factor $36$ out of $216$ .

$f (\sqrt{6}) = \frac{\sqrt{36 (6)}}{\sqrt{{\sqrt{6}}^{2} - 4}}$

Step 12.2.1.3.2

Rewrite $36$ as $6^{2}$ .

$f (\sqrt{6}) = \frac{\sqrt{6^{2} \cdot 6}}{\sqrt{{\sqrt{6}}^{2} - 4}}$

Step 12.2.1.4

Pull terms out from under the radical.

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{{\sqrt{6}}^{2} - 4}}$

Step 12.2.2

Simplify the denominator.

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Step 12.2.2.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 12.2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{{(6^{\frac{1}{2}})}^{2} - 4}}$

Step 12.2.2.1.2

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

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{6^{\frac{1}{2} \cdot 2} - 4}}$

Step 12.2.2.1.3

Combine $\frac{1}{2}$ and $2$ .

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{6^{\frac{2}{2}} - 4}}$

Step 12.2.2.1.4

Cancel the common factor of $2$ .

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Step 12.2.2.1.4.1

Cancel the common factor.

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{6^{\frac{2}{2}} - 4}}$

Step 12.2.2.1.4.2

Rewrite the expression.

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{6 - 4}}$

Step 12.2.2.1.5

Evaluate the exponent.

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{6 - 4}}$

Step 12.2.2.2

Subtract $4$ from $6$ .

$f (\sqrt{6}) = \frac{6 \sqrt{6}}{\sqrt{2}}$

Step 12.2.3

Combine $\sqrt{6}$ and $\sqrt{2}$ into a single radical.

$f (\sqrt{6}) = 6 \sqrt{\frac{6}{2}}$

Step 12.2.4

Divide $6$ by $2$ .

$f (\sqrt{6}) = 6 \sqrt{3}$

Step 12.2.5

The final answer is $6 \sqrt{3}$ .

$y = 6 \sqrt{3}$

Step 13

Evaluate the second derivative at $x = - \sqrt{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (- \sqrt{6}) ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{({(- \sqrt{6})}^{2} - 4)}^{\frac{5}{2}}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Multiply $- 1$ by $2$ .

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{({(- \sqrt{6})}^{2} - 4)}^{\frac{5}{2}}}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

Simplify each term.

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Step 14.2.1.1

Apply the product rule to $- \sqrt{6}$ .

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{({(- 1)}^{2} {\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.2

Raise $- 1$ to the power of $2$ .

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{(1 {\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.3

Multiply ${\sqrt{6}}^{2}$ by $1$ .

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{({\sqrt{6}}^{2} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.4

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 14.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{({(6^{\frac{1}{2}})}^{2} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.4.2

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

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{(6^{\frac{1}{2} \cdot 2} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{(6^{\frac{2}{2}} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.4.4

Cancel the common factor of $2$ .

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Step 14.2.1.4.4.1

Cancel the common factor.

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{(6^{\frac{2}{2}} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.4.4.2

Rewrite the expression.

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{(6^{1} - 4)}^{\frac{5}{2}}}$

Step 14.2.1.4.5

Evaluate the exponent.

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{{(6 - 4)}^{\frac{5}{2}}}$

Step 14.2.2

Subtract $4$ from $6$ .

$\frac{- 2 \sqrt{6} ({(- \sqrt{6})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3

Simplify the numerator.

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Step 14.3.1

Apply the product rule to $- \sqrt{6}$ .

$\frac{- 2 \sqrt{6} ({(- 1)}^{4} {\sqrt{6}}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.2

Raise $- 1$ to the power of $4$ .

$\frac{- 2 \sqrt{6} (1 {\sqrt{6}}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.3

Multiply ${\sqrt{6}}^{4}$ by $1$ .

$\frac{- 2 \sqrt{6} ({\sqrt{6}}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4

Rewrite ${\sqrt{6}}^{4}$ as $6^{2}$ .

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Step 14.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{- 2 \sqrt{6} ({(6^{\frac{1}{2}})}^{4} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4.2

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

$\frac{- 2 \sqrt{6} (6^{\frac{1}{2} \cdot 4} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{- 2 \sqrt{6} (6^{\frac{4}{2}} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4.4

Cancel the common factor of $4$ and $2$ .

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Step 14.3.4.4.1

Factor $2$ out of $4$ .

$\frac{- 2 \sqrt{6} (6^{\frac{2 \cdot 2}{2}} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4.4.2

Cancel the common factors.

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Step 14.3.4.4.2.1

Factor $2$ out of $2$ .

$\frac{- 2 \sqrt{6} (6^{\frac{2 \cdot 2}{2 (1)}} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4.4.2.2

Cancel the common factor.

$\frac{- 2 \sqrt{6} (6^{\frac{2 \cdot 2}{2 \cdot 1}} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4.4.2.3

Rewrite the expression.

$\frac{- 2 \sqrt{6} (6^{\frac{2}{1}} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.4.4.2.4

Divide $2$ by $1$ .

$\frac{- 2 \sqrt{6} (6^{2} - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.5

Raise $6$ to the power of $2$ .

$\frac{- 2 \sqrt{6} (36 - 10 {(- \sqrt{6})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.6

Apply the product rule to $- \sqrt{6}$ .

$\frac{- 2 \sqrt{6} (36 - 10 ({(- 1)}^{2} {\sqrt{6}}^{2}) + 48)}{2^{\frac{5}{2}}}$

Step 14.3.7

Raise $- 1$ to the power of $2$ .

$\frac{- 2 \sqrt{6} (36 - 10 (1 {\sqrt{6}}^{2}) + 48)}{2^{\frac{5}{2}}}$

Step 14.3.8

Multiply ${\sqrt{6}}^{2}$ by $1$ .

$\frac{- 2 \sqrt{6} (36 - 10 {\sqrt{6}}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.9

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 14.3.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{- 2 \sqrt{6} (36 - 10 {(6^{\frac{1}{2}})}^{2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.9.2

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

$\frac{- 2 \sqrt{6} (36 - 10 \cdot 6^{\frac{1}{2} \cdot 2} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.9.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- 2 \sqrt{6} (36 - 10 \cdot 6^{\frac{2}{2}} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.9.4

Cancel the common factor of $2$ .

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Step 14.3.9.4.1

Cancel the common factor.

$\frac{- 2 \sqrt{6} (36 - 10 \cdot 6^{\frac{2}{2}} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.9.4.2

Rewrite the expression.

$\frac{- 2 \sqrt{6} (36 - 10 \cdot 6^{1} + 48)}{2^{\frac{5}{2}}}$

Step 14.3.9.5

Evaluate the exponent.

$\frac{- 2 \sqrt{6} (36 - 10 \cdot 6 + 48)}{2^{\frac{5}{2}}}$

Step 14.3.10

Multiply $- 10$ by $6$ .

$\frac{- 2 \sqrt{6} (36 - 60 + 48)}{2^{\frac{5}{2}}}$

Step 14.3.11

Subtract $60$ from $36$ .

$\frac{- 2 \sqrt{6} (- 24 + 48)}{2^{\frac{5}{2}}}$

Step 14.3.12

Add $- 24$ and $48$ .

$\frac{- 2 \sqrt{6} \cdot 24}{2^{\frac{5}{2}}}$

Step 14.3.13

Multiply $24$ by $- 2$ .

$\frac{- 48 \sqrt{6}}{2^{\frac{5}{2}}}$

Step 14.4

Move the negative in front of the fraction.

$- \frac{48 \sqrt{6}}{2^{\frac{5}{2}}}$

Step 15

$x = - \sqrt{6}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \sqrt{6}$ is a local maximum

Step 16

Find the y-value when $x = - \sqrt{6}$ .

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Step 16.1

Replace the variable $x$ with $- \sqrt{6}$ in the expression.

$f (- \sqrt{6}) = \frac{{(- \sqrt{6})}^{3}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify the numerator.

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Step 16.2.1.1

Apply the product rule to $- \sqrt{6}$ .

$f (- \sqrt{6}) = \frac{{(- 1)}^{3} {\sqrt{6}}^{3}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.1.2

Raise $- 1$ to the power of $3$ .

$f (- \sqrt{6}) = \frac{- {\sqrt{6}}^{3}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.1.3

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$f (- \sqrt{6}) = \frac{- \sqrt{6^{3}}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.1.4

Raise $6$ to the power of $3$ .

$f (- \sqrt{6}) = \frac{- \sqrt{216}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.1.5

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 16.2.1.5.1

Factor $36$ out of $216$ .

$f (- \sqrt{6}) = \frac{- \sqrt{36 (6)}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.1.5.2

Rewrite $36$ as $6^{2}$ .

$f (- \sqrt{6}) = \frac{- \sqrt{6^{2} \cdot 6}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.1.6

Pull terms out from under the radical.

$f (- \sqrt{6}) = \frac{- 1 \cdot (6 \sqrt{6})}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.1.7

Multiply $- 1$ by $6$ .

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{{(- \sqrt{6})}^{2} - 4}}$

Step 16.2.2

Simplify the denominator.

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Step 16.2.2.1

Apply the product rule to $- \sqrt{6}$ .

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{{(- 1)}^{2} {\sqrt{6}}^{2} - 4}}$

Step 16.2.2.2

Raise $- 1$ to the power of $2$ .

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{1 {\sqrt{6}}^{2} - 4}}$

Step 16.2.2.3

Multiply ${\sqrt{6}}^{2}$ by $1$ .

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{{\sqrt{6}}^{2} - 4}}$

Step 16.2.2.4

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 16.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{{(6^{\frac{1}{2}})}^{2} - 4}}$

Step 16.2.2.4.2

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

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{6^{\frac{1}{2} \cdot 2} - 4}}$

Step 16.2.2.4.3

Combine $\frac{1}{2}$ and $2$ .

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{6^{\frac{2}{2}} - 4}}$

Step 16.2.2.4.4

Cancel the common factor of $2$ .

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Step 16.2.2.4.4.1

Cancel the common factor.

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{6^{\frac{2}{2}} - 4}}$

Step 16.2.2.4.4.2

Rewrite the expression.

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{6 - 4}}$

Step 16.2.2.4.5

Evaluate the exponent.

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{6 - 4}}$

Step 16.2.2.5

Subtract $4$ from $6$ .

$f (- \sqrt{6}) = \frac{- 6 \sqrt{6}}{\sqrt{2}}$

Step 16.2.3

Combine $\sqrt{6}$ and $\sqrt{2}$ into a single radical.

$f (- \sqrt{6}) = - 6 \sqrt{\frac{6}{2}}$

Step 16.2.4

Divide $6$ by $2$ .

$f (- \sqrt{6}) = - 6 \sqrt{3}$

Step 16.2.5

The final answer is $- 6 \sqrt{3}$ .

$y = - 6 \sqrt{3}$

Step 17

These are the local extrema for $f (x) = \frac{x^{3}}{\sqrt{x^{2} - 4}}$ .

$(\sqrt{6}, 6 \sqrt{3})$ is a local minima

$(- \sqrt{6}, - 6 \sqrt{3})$ is a local maxima

Step 18