Calculus Examples

$| 8 - x^{2} |$

Step 1

Write $| 8 - x^{2} |$ as a function.

$f (x) = | 8 - x^{2} |$

Step 2

Find the first derivative of the function.

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $8 - x^{2}$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [8 - x^{2}]$

Step 2.1.2

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

$\frac{u}{| u |} \frac{d}{d x} [8 - x^{2}]$

Step 2.1.3

Replace all occurrences of $u$ with $8 - x^{2}$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} \frac{d}{d x} [8 - x^{2}]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $8 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}]$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} (\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}])$

Step 2.2.2

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} (0 + \frac{d}{d x} [- x^{2}])$

Step 2.2.3

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

$\frac{8 - x^{2}}{| 8 - x^{2} |} \frac{d}{d x} [- x^{2}]$

Step 2.2.4

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

$\frac{8 - x^{2}}{| 8 - x^{2} |} (- \frac{d}{d x} [x^{2}])$

Step 2.2.5

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

$\frac{8 - x^{2}}{| 8 - x^{2} |} (- (2 x))$

Step 2.2.6

Combine fractions.

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Step 2.2.6.1

Multiply $2$ by $- 1$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} (- 2 x)$

Step 2.2.6.2

Combine $- 2$ and $\frac{8 - x^{2}}{| 8 - x^{2} |}$ .

$\frac{- 2 (8 - x^{2})}{| 8 - x^{2} |} x$

Step 2.2.6.3

Combine $\frac{- 2 (8 - x^{2})}{| 8 - x^{2} |}$ and $x$ .

$\frac{- 2 (8 - x^{2}) x}{| 8 - x^{2} |}$

Step 2.2.6.4

Move the negative in front of the fraction.

$- \frac{(2 (8 - x^{2})) x}{| 8 - x^{2} |}$

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$- \frac{(2 \cdot 8 + 2 (- x^{2})) x}{| 8 - x^{2} |}$

Step 2.3.2

Apply the distributive property.

$- \frac{2 \cdot 8 x + 2 (- x^{2}) x}{| 8 - x^{2} |}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Multiply $2$ by $8$ .

$- \frac{16 x + 2 (- x^{2}) x}{| 8 - x^{2} |}$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Move $x$ .

$- \frac{16 x + 2 (- (x \cdot x^{2}))}{| 8 - x^{2} |}$

Step 2.3.3.2.2

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

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Step 2.3.3.2.2.1

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

$- \frac{16 x + 2 (- (x^{1} x^{2}))}{| 8 - x^{2} |}$

Step 2.3.3.2.2.2

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

$- \frac{16 x + 2 (- x^{1 + 2})}{| 8 - x^{2} |}$

Step 2.3.3.2.3

Add $1$ and $2$ .

$- \frac{16 x + 2 (- x^{3})}{| 8 - x^{2} |}$

Step 2.3.3.3

Multiply $- 1$ by $2$ .

$- \frac{16 x - 2 x^{3}}{| 8 - x^{2} |}$

Step 2.3.4

Factor $2 x$ out of $16 x - 2 x^{3}$ .

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Step 2.3.4.1

Factor $2 x$ out of $16 x$ .

$- \frac{2 x (8) - 2 x^{3}}{| 8 - x^{2} |}$

Step 2.3.4.2

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

$- \frac{2 x (8) + 2 x (- x^{2})}{| 8 - x^{2} |}$

Step 2.3.4.3

Factor $2 x$ out of $2 x (8) + 2 x (- x^{2})$ .

$- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$

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 $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{x (8 - x^{2})}{| 8 - x^{2} |}]$ .

$f'' (x) = - 2 \frac{d}{d x} \frac{x (8 - x^{2})}{| 8 - x^{2} |}$

Step 3.2

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 $f (x) = x (8 - x^{2})$ and $g (x) = | 8 - x^{2} |$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | \frac{d}{d x} (x (8 - x^{2})) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.3

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 $g (x) = 8 - x^{2}$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (x \frac{d}{d x} (8 - x^{2}) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $8 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}]$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (x (\frac{d}{d x} (8) + \frac{d}{d x} (- x^{2})) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.4.2

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (x (0 + \frac{d}{d x} (- x^{2})) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.4.3

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (x \frac{d}{d x} (- x^{2}) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.4.4

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (x (- \frac{d}{d x} x^{2}) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.4.5

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (x (- (2 x)) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.4.6

Multiply $2$ by $- 1$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (x (- 2 x) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.5

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 2 (x \cdot x) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.6

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 2 (x \cdot x) + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.7

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 2 x^{1 + 1} + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.8

Differentiate using the Power Rule.

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Step 3.8.1

Add $1$ and $1$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 2 x^{2} + (8 - x^{2}) \frac{d}{d x} (x)) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.8.2

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 2 x^{2} + (8 - x^{2}) \cdot 1) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.8.3

Simplify by adding terms.

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Step 3.8.3.1

Multiply $8 - x^{2}$ by $1$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 2 x^{2} + 8 - x^{2}) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.8.3.2

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - x (8 - x^{2}) \frac{d}{d x} | 8 - x^{2} |}{{| 8 - x^{2} |}^{2}}$

Step 3.9

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

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Step 3.9.1

To apply the Chain Rule, set $u$ as $8 - x^{2}$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - x (8 - x^{2}) (\frac{d}{d u} (| u |) \frac{d}{d x} (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.9.2

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - x (8 - x^{2}) (\frac{u}{| u |} \cdot \frac{d}{d x} (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.9.3

Replace all occurrences of $u$ with $8 - x^{2}$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - x (8 - x^{2}) (\frac{8 - x^{2}}{| 8 - x^{2} |} \cdot \frac{d}{d x} (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.10

Differentiate.

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Step 3.10.1

Combine $\frac{8 - x^{2}}{| 8 - x^{2} |}$ and $x$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - \frac{(8 - x^{2}) x}{| 8 - x^{2} |} \cdot (8 - x^{2}) \frac{d}{d x} (8 - x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.10.2

By the Sum Rule, the derivative of $8 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}]$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - \frac{(8 - x^{2}) x}{| 8 - x^{2} |} \cdot ((8 - x^{2}) (\frac{d}{d x} (8) + \frac{d}{d x} (- x^{2})))}{{| 8 - x^{2} |}^{2}}$

Step 3.10.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - \frac{(8 - x^{2}) x}{| 8 - x^{2} |} \cdot ((8 - x^{2}) (0 + \frac{d}{d x} (- x^{2})))}{{| 8 - x^{2} |}^{2}}$

Step 3.10.4

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - \frac{(8 - x^{2}) x}{| 8 - x^{2} |} \cdot (8 - x^{2}) \frac{d}{d x} (- x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.10.5

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) - \frac{(8 - x^{2}) x}{| 8 - x^{2} |} \cdot ((8 - x^{2}) (- \frac{d}{d x} x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.10.6

Multiply.

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Step 3.10.6.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + 1 (\frac{(8 - x^{2}) x}{| 8 - x^{2} |}) \cdot (8 - x^{2}) \frac{d}{d x} (x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.10.6.2

Multiply $\frac{(8 - x^{2}) x}{| 8 - x^{2} |}$ by $1$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{(8 - x^{2}) x}{| 8 - x^{2} |} \cdot (8 - x^{2}) \frac{d}{d x} (x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.10.7

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{(8 - x^{2}) x}{| 8 - x^{2} |} \cdot ((8 - x^{2}) (2 x))}{{| 8 - x^{2} |}^{2}}$

Step 3.10.8

Combine fractions.

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Step 3.10.8.1

Combine $2$ and $\frac{(8 - x^{2}) x}{| 8 - x^{2} |}$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{2 ((8 - x^{2}) x)}{| 8 - x^{2} |} \cdot ((8 - x^{2}) x)}{{| 8 - x^{2} |}^{2}}$

Step 3.10.8.2

Combine $x$ and $\frac{2 ((8 - x^{2}) x)}{| 8 - x^{2} |}$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x (2 ((8 - x^{2}) x))}{| 8 - x^{2} |} \cdot (8 - x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.11

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x \cdot x (2 (8 - x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.12

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x \cdot x (2 (8 - x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.13

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

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x^{1 + 1} (2 (8 - x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.14

Add $1$ and $1$ .

$f'' (x) = - 2 \frac{| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x^{2} (2 (8 - x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2})}{{| 8 - x^{2} |}^{2}}$

Step 3.15

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

$f'' (x) = \frac{- 2 (| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x^{2} (2 (8 - x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.16

Move the negative in front of the fraction.

$f'' (x) = - \frac{2 (| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x^{2} (2 (8 - x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17

Simplify.

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Step 3.17.1

Apply the distributive property.

$f'' (x) = - \frac{2 (| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x^{2} (2 \cdot 8 + 2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.2

Apply the distributive property.

$f'' (x) = - \frac{2 (| 8 - x^{2} | (- 3 x^{2} + 8) + \frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.3

Apply the distributive property.

$f'' (x) = - \frac{2 (| 8 - x^{2} | (- 3 x^{2} + 8)) + 2 (\frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4

Simplify the numerator.

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Step 3.17.4.1

Simplify each term.

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Step 3.17.4.1.1

Apply the distributive property.

$f'' (x) = - \frac{2 (| 8 - x^{2} | (- 3 x^{2}) + | 8 - x^{2} | \cdot 8) + 2 (\frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{2 (- 3 | 8 - x^{2} | x^{2} + | 8 - x^{2} | \cdot 8) + 2 (\frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.3

Move $8$ to the left of $| 8 - x^{2} |$ .

$f'' (x) = - \frac{2 (- 3 | 8 - x^{2} | x^{2} + 8 \cdot | 8 - x^{2} |) + 2 (\frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.4

Apply the distributive property.

$f'' (x) = - \frac{2 (- 3 | 8 - x^{2} | x^{2}) + 2 (8 | 8 - x^{2} |) + 2 (\frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.5

Multiply $- 3$ by $2$ .

$f'' (x) = - \frac{- 6 (| 8 - x^{2} | x^{2}) + 2 (8 | 8 - x^{2} |) + 2 (\frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.6

Multiply $8$ by $2$ .

$f'' (x) = - \frac{- 6 (| 8 - x^{2} | x^{2}) + 16 | 8 - x^{2} | + 2 (\frac{x^{2} (2 \cdot 8) + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7

Simplify the numerator.

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Step 3.17.4.1.7.1

Multiply $2$ by $8$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{x^{2} \cdot 16 + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.2

Move $16$ to the left of $x^{2}$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{16 \cdot x^{2} + x^{2} (2 (- x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{16 x^{2} + 2 \cdot (- 1 x^{2} x^{2})}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.4

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

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Step 3.17.4.1.7.4.1

Move $x^{2}$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{16 x^{2} + 2 \cdot (- 1 (x^{2} x^{2}))}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.4.2

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

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{16 x^{2} + 2 \cdot (- 1 x^{2 + 2})}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.4.3

Add $2$ and $2$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{16 x^{2} + 2 \cdot (- 1 x^{4})}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.5

Multiply $2$ by $- 1$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{16 x^{2} - 2 x^{4}}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.6

Factor $2 x^{2}$ out of $16 x^{2} - 2 x^{4}$ .

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Step 3.17.4.1.7.6.1

Factor $2 x^{2}$ out of $16 x^{2}$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} (8) - 2 x^{4}}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.6.2

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

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} (8) + 2 x^{2} (- x^{2})}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.7.6.3

Factor $2 x^{2}$ out of $2 x^{2} (8) + 2 x^{2} (- x^{2})$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} (8 - x^{2})}{| 8 - x^{2} |} \cdot (8 - x^{2}))}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.8

Multiply $\frac{2 x^{2} (8 - x^{2})}{| 8 - x^{2} |}$ by $8 - x^{2}$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} (8 - x^{2}) (8 - x^{2})}{| 8 - x^{2} |})}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.9

Simplify the numerator.

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Step 3.17.4.1.9.1

Raise $8 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} ((8 - x^{2}) (8 - x^{2}))}{| 8 - x^{2} |})}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.9.2

Raise $8 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} ((8 - x^{2}) (8 - x^{2}))}{| 8 - x^{2} |})}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.9.3

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

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} {(8 - x^{2})}^{1 + 1}}{| 8 - x^{2} |})}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.9.4

Add $1$ and $1$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + 2 (\frac{2 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |})}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.10

Multiply $2 \frac{2 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |}$ .

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Step 3.17.4.1.10.1

Combine $2$ and $\frac{2 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |}$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + \frac{2 (2 x^{2} {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.1.10.2

Multiply $2$ by $2$ .

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + \frac{4 (x^{2} {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

$f'' (x) = - \frac{- 6 | 8 - x^{2} | x^{2} + 16 | 8 - x^{2} | + \frac{4 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.2

To write $- 6 | 8 - x^{2} | x^{2}$ as a fraction with a common denominator, multiply by $\frac{| 8 - x^{2} |}{| 8 - x^{2} |}$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | - 6 | 8 - x^{2} | x^{2} \cdot \frac{| 8 - x^{2} |}{| 8 - x^{2} |} + \frac{4 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.3

Combine $- 6 | 8 - x^{2} | x^{2}$ and $\frac{| 8 - x^{2} |}{| 8 - x^{2} |}$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{- 6 | 8 - x^{2} | x^{2} | 8 - x^{2} |}{| 8 - x^{2} |} + \frac{4 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{- 6 | 8 - x^{2} | x^{2} | 8 - x^{2} | + 4 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5

Simplify the numerator.

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Step 3.17.4.5.1

Factor $2 x^{2}$ out of $- 6 | 8 - x^{2} | x^{2} | 8 - x^{2} | + 4 x^{2} {(8 - x^{2})}^{2}$ .

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Step 3.17.4.5.1.1

Factor $2 x^{2}$ out of $- 6 | 8 - x^{2} | x^{2} | 8 - x^{2} |$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 8 - x^{2} | | 8 - x^{2} |) + 4 x^{2} {(8 - x^{2})}^{2}}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.1.2

Factor $2 x^{2}$ out of $4 x^{2} {(8 - x^{2})}^{2}$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 8 - x^{2} | | 8 - x^{2} |) + 2 x^{2} (2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.1.3

Factor $2 x^{2}$ out of $2 x^{2} (- 3 | 8 - x^{2} | | 8 - x^{2} |) + 2 x^{2} (2 {(8 - x^{2})}^{2})$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 8 - x^{2} | | 8 - x^{2} | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.2

Combine exponents.

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Step 3.17.4.5.2.1

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | (8 - x^{2}) (8 - x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.2.2

Raise $8 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | (8 - x^{2}) (8 - x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.2.3

Raise $8 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | (8 - x^{2}) (8 - x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.2.4

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

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | {(8 - x^{2})}^{1 + 1} | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.2.5

Add $1$ and $1$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | {(8 - x^{2})}^{2} | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3

Simplify each term.

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Step 3.17.4.5.3.1

Rewrite ${(8 - x^{2})}^{2}$ as $(8 - x^{2}) (8 - x^{2})$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | (8 - x^{2}) (8 - x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.2

Expand $(8 - x^{2}) (8 - x^{2})$ using the FOIL Method.

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Step 3.17.4.5.3.2.1

Apply the distributive property.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 8 (8 - x^{2}) - x^{2} (8 - x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.2.2

Apply the distributive property.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 8 \cdot 8 + 8 (- x^{2}) - x^{2} (8 - x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.2.3

Apply the distributive property.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 8 \cdot 8 + 8 (- x^{2}) - x^{2} \cdot 8 - x^{2} (- x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3

Simplify and combine like terms.

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Step 3.17.4.5.3.3.1

Simplify each term.

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Step 3.17.4.5.3.3.1.1

Multiply $8$ by $8$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 + 8 (- x^{2}) - x^{2} \cdot 8 - x^{2} (- x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.2

Multiply $- 1$ by $8$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - x^{2} \cdot 8 - x^{2} (- x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.3

Multiply $8$ by $- 1$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - 8 x^{2} - x^{2} (- x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{2} x^{2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.5

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

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Step 3.17.4.5.3.3.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 (x^{2} x^{2})) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.5.2

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

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{2 + 2}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{4}) | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - 8 x^{2} + 1 x^{4} | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.1.7

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

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 8 x^{2} - 8 x^{2} + x^{4} | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.3.2

Subtract $8 x^{2}$ from $- 8 x^{2}$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 {(8 - x^{2})}^{2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.4

Rewrite ${(8 - x^{2})}^{2}$ as $(8 - x^{2}) (8 - x^{2})$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 ((8 - x^{2}) (8 - x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.5

Expand $(8 - x^{2}) (8 - x^{2})$ using the FOIL Method.

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Step 3.17.4.5.3.5.1

Apply the distributive property.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (8 (8 - x^{2}) - x^{2} (8 - x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.5.2

Apply the distributive property.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (8 \cdot 8 + 8 (- x^{2}) - x^{2} (8 - x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.5.3

Apply the distributive property.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (8 \cdot 8 + 8 (- x^{2}) - x^{2} \cdot 8 - x^{2} (- x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6

Simplify and combine like terms.

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Step 3.17.4.5.3.6.1

Simplify each term.

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Step 3.17.4.5.3.6.1.1

Multiply $8$ by $8$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 + 8 (- x^{2}) - x^{2} \cdot 8 - x^{2} (- x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.2

Multiply $- 1$ by $8$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - x^{2} \cdot 8 - x^{2} (- x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.3

Multiply $8$ by $- 1$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - 8 x^{2} - x^{2} (- x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{2} x^{2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.5

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

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Step 3.17.4.5.3.6.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 (x^{2} x^{2}))))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.5.2

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

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{2 + 2})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{4})))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - 8 x^{2} + 1 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.1.7

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

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 8 x^{2} - 8 x^{2} + x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.6.2

Subtract $8 x^{2}$ from $- 8 x^{2}$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 (64 - 16 x^{2} + x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.7

Apply the distributive property.

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 2 \cdot 64 + 2 (- 16 x^{2}) + 2 x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.8

Simplify.

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Step 3.17.4.5.3.8.1

Multiply $2$ by $64$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 + 2 (- 16 x^{2}) + 2 x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.5.3.8.2

Multiply $- 16$ by $2$ .

$f'' (x) = - \frac{16 | 8 - x^{2} | + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.6

To write $16 | 8 - x^{2} |$ as a fraction with a common denominator, multiply by $\frac{| 8 - x^{2} |}{| 8 - x^{2} |}$ .

$f'' (x) = - \frac{\frac{16 | 8 - x^{2} | | 8 - x^{2} |}{| 8 - x^{2} |} + \frac{2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.7

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{16 | 8 - x^{2} | | 8 - x^{2} | + 2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8

Simplify the numerator.

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Step 3.17.4.8.1

Factor $2$ out of $16 | 8 - x^{2} | | 8 - x^{2} | + 2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4})$ .

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Step 3.17.4.8.1.1

Factor $2$ out of $16 | 8 - x^{2} | | 8 - x^{2} |$ .

$f'' (x) = - \frac{\frac{2 (8 | 8 - x^{2} | | 8 - x^{2} |) + 2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.1.2

Factor $2$ out of $2 x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4})$ .

$f'' (x) = - \frac{\frac{2 (8 | 8 - x^{2} | | 8 - x^{2} |) + 2 (x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.1.3

Factor $2$ out of $2 (8 | 8 - x^{2} | | 8 - x^{2} |) + 2 (x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))$ .

$f'' (x) = - \frac{\frac{2 (8 | 8 - x^{2} | | 8 - x^{2} | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.2

Multiply $8 | 8 - x^{2} | | 8 - x^{2} |$ .

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Step 3.17.4.8.2.1

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = - \frac{\frac{2 (8 | (8 - x^{2}) (8 - x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.2.2

Raise $8 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{\frac{2 (8 | (8 - x^{2}) (8 - x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.2.3

Raise $8 - x^{2}$ to the power of $1$ .

$f'' (x) = - \frac{\frac{2 (8 | (8 - x^{2}) (8 - x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.2.4

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

$f'' (x) = - \frac{\frac{2 (8 | {(8 - x^{2})}^{1 + 1} | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.2.5

Add $1$ and $1$ .

$f'' (x) = - \frac{\frac{2 (8 | {(8 - x^{2})}^{2} | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.3

Rewrite ${(8 - x^{2})}^{2}$ as $(8 - x^{2}) (8 - x^{2})$ .

$f'' (x) = - \frac{\frac{2 (8 | (8 - x^{2}) (8 - x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.4

Expand $(8 - x^{2}) (8 - x^{2})$ using the FOIL Method.

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Step 3.17.4.8.4.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (8 | 8 (8 - x^{2}) - x^{2} (8 - x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.4.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (8 | 8 \cdot 8 + 8 (- x^{2}) - x^{2} (8 - x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.4.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (8 | 8 \cdot 8 + 8 (- x^{2}) - x^{2} \cdot 8 - x^{2} (- x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5

Simplify and combine like terms.

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Step 3.17.4.8.5.1

Simplify each term.

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Step 3.17.4.8.5.1.1

Multiply $8$ by $8$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 + 8 (- x^{2}) - x^{2} \cdot 8 - x^{2} (- x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.2

Multiply $- 1$ by $8$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - x^{2} \cdot 8 - x^{2} (- x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.3

Multiply $8$ by $- 1$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - 8 x^{2} - x^{2} (- x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{2} x^{2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.5

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

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Step 3.17.4.8.5.1.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 (x^{2} x^{2})) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.5.2

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

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{2 + 2}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.5.3

Add $2$ and $2$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - 8 x^{2} - 1 \cdot (- 1 x^{4}) | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.6

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - 8 x^{2} + 1 x^{4} | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.1.7

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

$f'' (x) = - \frac{\frac{2 (8 | 64 - 8 x^{2} - 8 x^{2} + x^{4} | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.5.2

Subtract $8 x^{2}$ from $- 8 x^{2}$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} | + 128 - 32 x^{2} + 2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.6

Apply the distributive property.

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | + x^{2} (- 3 | 64 - 16 x^{2} + x^{4} |) + x^{2} \cdot 128 + x^{2} (- 32 x^{2}) + x^{2} (2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.7

Simplify.

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Step 3.17.4.8.7.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + x^{2} \cdot 128 + x^{2} (- 32 x^{2}) + x^{2} (2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.7.2

Move $128$ to the left of $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 \cdot x^{2} + x^{2} (- 32 x^{2}) + x^{2} (2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.7.3

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 \cdot x^{2} - 32 x^{2} x^{2} + x^{2} (2 x^{4}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.7.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 \cdot x^{2} - 32 x^{2} x^{2} + 2 x^{2} x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.8

Simplify each term.

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Step 3.17.4.8.8.1

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

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Step 3.17.4.8.8.1.1

Move $x^{2}$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 (x^{2} x^{2}) + 2 x^{2} x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.8.1.2

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

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{2 + 2} + 2 x^{2} x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.8.1.3

Add $2$ and $2$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{2} x^{4})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.8.2

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

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Step 3.17.4.8.8.2.1

Move $x^{4}$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 (x^{4} x^{2}))}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.8.2.2

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

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{4 + 2})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.4.8.8.2.3

Add $4$ and $2$ .

$f'' (x) = - \frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$

Step 3.17.5

Combine terms.

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Step 3.17.5.1

Rewrite $\frac{\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{| 8 - x^{2} |}}{{| 8 - x^{2} |}^{2}}$ as a product.

$f'' (x) = - (\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{| 8 - x^{2} |} \cdot \frac{1}{{| 8 - x^{2} |}^{2}})$

Step 3.17.5.2

Multiply $\frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{| 8 - x^{2} |}$ by $\frac{1}{{| 8 - x^{2} |}^{2}}$ .

$f'' (x) = - \frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{| 8 - x^{2} | {| 8 - x^{2} |}^{2}}$

Step 3.17.5.3

Multiply $| 8 - x^{2} |$ by ${| 8 - x^{2} |}^{2}$ by adding the exponents.

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Step 3.17.5.3.1

Multiply $| 8 - x^{2} |$ by ${| 8 - x^{2} |}^{2}$ .

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Step 3.17.5.3.1.1

Raise $| 8 - x^{2} |$ to the power of $1$ .

$f'' (x) = - \frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{| 8 - x^{2} | {| 8 - x^{2} |}^{2}}$

Step 3.17.5.3.1.2

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

$f'' (x) = - \frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{{| 8 - x^{2} |}^{1 + 2}}$

Step 3.17.5.3.2

Add $1$ and $2$ .

$f'' (x) = - \frac{2 (8 | 64 - 16 x^{2} + x^{4} | - 3 x^{2} | 64 - 16 x^{2} + x^{4} | + 128 x^{2} - 32 x^{4} + 2 x^{6})}{{| 8 - x^{2} |}^{3}}$

Step 4

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

$- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |} = 0$

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

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

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Step 5.1.1.1

To apply the Chain Rule, set $u$ as $8 - x^{2}$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [8 - x^{2}]$

Step 5.1.1.2

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

$\frac{u}{| u |} \frac{d}{d x} [8 - x^{2}]$

Step 5.1.1.3

Replace all occurrences of $u$ with $8 - x^{2}$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} \frac{d}{d x} [8 - x^{2}]$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $8 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}]$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} (\frac{d}{d x} [8] + \frac{d}{d x} [- x^{2}])$

Step 5.1.2.2

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} (0 + \frac{d}{d x} [- x^{2}])$

Step 5.1.2.3

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

$\frac{8 - x^{2}}{| 8 - x^{2} |} \frac{d}{d x} [- x^{2}]$

Step 5.1.2.4

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

$\frac{8 - x^{2}}{| 8 - x^{2} |} (- \frac{d}{d x} [x^{2}])$

Step 5.1.2.5

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

$\frac{8 - x^{2}}{| 8 - x^{2} |} (- (2 x))$

Step 5.1.2.6

Combine fractions.

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Step 5.1.2.6.1

Multiply $2$ by $- 1$ .

$\frac{8 - x^{2}}{| 8 - x^{2} |} (- 2 x)$

Step 5.1.2.6.2

Combine $- 2$ and $\frac{8 - x^{2}}{| 8 - x^{2} |}$ .

$\frac{- 2 (8 - x^{2})}{| 8 - x^{2} |} x$

Step 5.1.2.6.3

Combine $\frac{- 2 (8 - x^{2})}{| 8 - x^{2} |}$ and $x$ .

$\frac{- 2 (8 - x^{2}) x}{| 8 - x^{2} |}$

Step 5.1.2.6.4

Move the negative in front of the fraction.

$- \frac{(2 (8 - x^{2})) x}{| 8 - x^{2} |}$

Step 5.1.3

Simplify.

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Step 5.1.3.1

Apply the distributive property.

$- \frac{(2 \cdot 8 + 2 (- x^{2})) x}{| 8 - x^{2} |}$

Step 5.1.3.2

Apply the distributive property.

$- \frac{2 \cdot 8 x + 2 (- x^{2}) x}{| 8 - x^{2} |}$

Step 5.1.3.3

Simplify each term.

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Step 5.1.3.3.1

Multiply $2$ by $8$ .

$- \frac{16 x + 2 (- x^{2}) x}{| 8 - x^{2} |}$

Step 5.1.3.3.2

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

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Step 5.1.3.3.2.1

Move $x$ .

$- \frac{16 x + 2 (- (x \cdot x^{2}))}{| 8 - x^{2} |}$

Step 5.1.3.3.2.2

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

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Step 5.1.3.3.2.2.1

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

$- \frac{16 x + 2 (- (x^{1} x^{2}))}{| 8 - x^{2} |}$

Step 5.1.3.3.2.2.2

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

$- \frac{16 x + 2 (- x^{1 + 2})}{| 8 - x^{2} |}$

Step 5.1.3.3.2.3

Add $1$ and $2$ .

$- \frac{16 x + 2 (- x^{3})}{| 8 - x^{2} |}$

Step 5.1.3.3.3

Multiply $- 1$ by $2$ .

$- \frac{16 x - 2 x^{3}}{| 8 - x^{2} |}$

Step 5.1.3.4

Factor $2 x$ out of $16 x - 2 x^{3}$ .

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Step 5.1.3.4.1

Factor $2 x$ out of $16 x$ .

$- \frac{2 x (8) - 2 x^{3}}{| 8 - x^{2} |}$

Step 5.1.3.4.2

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

$- \frac{2 x (8) + 2 x (- x^{2})}{| 8 - x^{2} |}$

Step 5.1.3.4.3

Factor $2 x$ out of $2 x (8) + 2 x (- x^{2})$ .

$f' (x) = - \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$ .

$- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |} = 0$

Step 6.2

Set the numerator equal to zero.

$2 x (8 - x^{2}) = 0$

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 $0$ .

$x = 0$

$8 - x^{2} = 0$

Step 6.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.3

Set $8 - x^{2}$ equal to $0$ and solve for $x$ .

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Step 6.3.3.1

Set $8 - x^{2}$ equal to $0$ .

$8 - x^{2} = 0$

Step 6.3.3.2

Solve $8 - x^{2} = 0$ for $x$ .

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Step 6.3.3.2.1

Subtract $8$ from both sides of the equation.

$- x^{2} = - 8$

Step 6.3.3.2.2

Divide each term in $- x^{2} = - 8$ by $- 1$ and simplify.

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Step 6.3.3.2.2.1

Divide each term in $- x^{2} = - 8$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 8}{- 1}$

Step 6.3.3.2.2.2

Simplify the left side.

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Step 6.3.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 8}{- 1}$

Step 6.3.3.2.2.2.2

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

$x^{2} = \frac{- 8}{- 1}$

Step 6.3.3.2.2.3

Simplify the right side.

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Step 6.3.3.2.2.3.1

Divide $- 8$ by $- 1$ .

$x^{2} = 8$

Step 6.3.3.2.3

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

$x = \pm \sqrt{8}$

Step 6.3.3.2.4

Simplify $\pm \sqrt{8}$ .

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Step 6.3.3.2.4.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.3.3.2.4.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 6.3.3.2.4.1.2

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

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

Step 6.3.3.2.4.2

Pull terms out from under the radical.

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

Step 6.3.3.2.5

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

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Step 6.3.3.2.5.1

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

$x = 2 \sqrt{2}$

Step 6.3.3.2.5.2

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

$x = - 2 \sqrt{2}$

Step 6.3.3.2.5.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 6.3.4

The final solution is all the values that make $2 x (8 - x^{2}) = 0$ true.

$x = 0, 2 \sqrt{2}, - 2 \sqrt{2}$

Step 6.4

Exclude the solutions that do not make $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |} = 0$ true.

$x = 0$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$ equal to $0$ to find where the expression is undefined.

$| 8 - x^{2} | = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$8 - x^{2} = \pm 0$

Step 7.2.2

Plus or minus $0$ is $0$ .

$8 - x^{2} = 0$

Step 7.2.3

Subtract $8$ from both sides of the equation.

$- x^{2} = - 8$

Step 7.2.4

Divide each term in $- x^{2} = - 8$ by $- 1$ and simplify.

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Step 7.2.4.1

Divide each term in $- x^{2} = - 8$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 8}{- 1}$

Step 7.2.4.2

Simplify the left side.

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Step 7.2.4.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 8}{- 1}$

Step 7.2.4.2.2

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

$x^{2} = \frac{- 8}{- 1}$

Step 7.2.4.3

Simplify the right side.

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Step 7.2.4.3.1

Divide $- 8$ by $- 1$ .

$x^{2} = 8$

Step 7.2.5

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

$x = \pm \sqrt{8}$

Step 7.2.6

Simplify $\pm \sqrt{8}$ .

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Step 7.2.6.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.2.6.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 7.2.6.1.2

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

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

Step 7.2.6.2

Pull terms out from under the radical.

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

Step 7.2.7

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

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Step 7.2.7.1

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

$x = 2 \sqrt{2}$

Step 7.2.7.2

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

$x = - 2 \sqrt{2}$

Step 7.2.7.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 7.3

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

$x = - 2 \sqrt{2}, x = 2 \sqrt{2}$

Step 8

Critical points to evaluate.

$x = 0, - 2 \sqrt{2}, 2 \sqrt{2}$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (8 | 64 - 16 {(0)}^{2} + {(0)}^{4} | - 3 {(0)}^{2} | 64 - 16 {(0)}^{2} + {(0)}^{4} | + 128 {(0)}^{2} - 32 {(0)}^{4} + 2 {(0)}^{6})}{{| 8 - {(0)}^{2} |}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Simplify each term.

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Step 10.1.1.1

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

$- \frac{2 (8 | 64 - 16 \cdot 0 + 0^{4} | - 3 \cdot 0^{2} | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.1.2

Multiply $- 16$ by $0$ .

$- \frac{2 (8 | 64 + 0 + 0^{4} | - 3 \cdot 0^{2} | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.1.3

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

$- \frac{2 (8 | 64 + 0 + 0 | - 3 \cdot 0^{2} | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.2

Add $64$ and $0$ .

$- \frac{2 (8 | 64 + 0 | - 3 \cdot 0^{2} | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.3

Add $64$ and $0$ .

$- \frac{2 (8 | 64 | - 3 \cdot 0^{2} | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.4

The absolute value is the distance between a number and zero. The distance between $0$ and $64$ is $64$ .

$- \frac{2 (8 \cdot 64 - 3 \cdot 0^{2} | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.5

Multiply $8$ by $64$ .

$- \frac{2 (512 - 3 \cdot 0^{2} | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.6

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

$- \frac{2 (512 - 3 \cdot 0 | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.7

Multiply $- 3$ by $0$ .

$- \frac{2 (512 + 0 | 64 - 16 \cdot 0^{2} + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.8

Simplify each term.

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Step 10.1.8.1

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

$- \frac{2 (512 + 0 | 64 - 16 \cdot 0 + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.8.2

Multiply $- 16$ by $0$ .

$- \frac{2 (512 + 0 | 64 + 0 + 0^{4} | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.8.3

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

$- \frac{2 (512 + 0 | 64 + 0 + 0 | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.9

Add $64$ and $0$ .

$- \frac{2 (512 + 0 | 64 + 0 | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.10

Add $64$ and $0$ .

$- \frac{2 (512 + 0 | 64 | + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.11

The absolute value is the distance between a number and zero. The distance between $0$ and $64$ is $64$ .

$- \frac{2 (512 + 0 \cdot 64 + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.12

Multiply $0$ by $64$ .

$- \frac{2 (512 + 0 + 128 \cdot 0^{2} - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.13

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

$- \frac{2 (512 + 0 + 128 \cdot 0 - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.14

Multiply $128$ by $0$ .

$- \frac{2 (512 + 0 + 0 - 32 \cdot 0^{4} + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.15

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

$- \frac{2 (512 + 0 + 0 - 32 \cdot 0 + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.16

Multiply $- 32$ by $0$ .

$- \frac{2 (512 + 0 + 0 + 0 + 2 \cdot 0^{6})}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.17

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

$- \frac{2 (512 + 0 + 0 + 0 + 2 \cdot 0)}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.18

Multiply $2$ by $0$ .

$- \frac{2 (512 + 0 + 0 + 0 + 0)}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.19

Add $512 + 0 + 0 + 0$ and $0$ .

$- \frac{2 (512 + 0 + 0 + 0)}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.20

Add $512 + 0 + 0$ and $0$ .

$- \frac{2 (512 + 0 + 0)}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.21

Add $512 + 0$ and $0$ .

$- \frac{2 (512 + 0)}{{| 8 - 0^{2} |}^{3}}$

Step 10.1.22

Add $512$ and $0$ .

$- \frac{2 \cdot 512}{{| 8 - 0^{2} |}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

$- \frac{2 \cdot 512}{{| 8 - 0 |}^{3}}$

Step 10.2.2

Multiply $- 1$ by $0$ .

$- \frac{2 \cdot 512}{{| 8 + 0 |}^{3}}$

Step 10.2.3

Add $8$ and $0$ .

$- \frac{2 \cdot 512}{{| 8 |}^{3}}$

Step 10.2.4

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$- \frac{2 \cdot 512}{8^{3}}$

Step 10.2.5

Raise $8$ to the power of $3$ .

$- \frac{2 \cdot 512}{512}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $2$ by $512$ .

$- \frac{1024}{512}$

Step 10.3.2

Divide $1024$ by $512$ .

$- 1 \cdot 2$

Step 10.3.3

Multiply $- 1$ by $2$ .

$- 2$

Step 11

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 12

Find the y-value when $x = 0$ .

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Step 12.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = | 8 - {(0)}^{2} |$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

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

$f (0) = | 8 - 0 |$

Step 12.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = | 8 + 0 |$

Step 12.2.2

Add $8$ and $0$ .

$f (0) = | 8 |$

Step 12.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $8$ is $8$ .

$f (0) = 8$

Step 12.2.4

The final answer is $8$ .

$y = 8$

Step 13

Evaluate the second derivative at $x = - 2 \sqrt{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (8 | 64 - 16 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{4} | - 3 {(- 2 \sqrt{2})}^{2} | 64 - 16 {(- 2 \sqrt{2})}^{2} + {(- 2 \sqrt{2})}^{4} | + 128 {(- 2 \sqrt{2})}^{2} - 32 {(- 2 \sqrt{2})}^{4} + 2 {(- 2 \sqrt{2})}^{6})}{{| 8 - {(- 2 \sqrt{2})}^{2} |}^{3}}$

Step 14

Evaluate the second derivative.

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Step 14.1

Simplify each term.

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Step 14.1.1

Apply the product rule to $- 2 \sqrt{2}$ .

Step 14.1.2

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

Step 14.1.3

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

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Step 14.1.3.1

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

Step 14.1.3.2

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

Step 14.1.3.3

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

Step 14.1.3.4

Cancel the common factor of $2$ .

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Step 14.1.3.4.1

Cancel the common factor.

Step 14.1.3.4.2

Rewrite the expression.

Step 14.1.3.5

Evaluate the exponent.

Step 14.1.4

Multiply $- (4 \cdot 2)$ .

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Step 14.1.4.1

Multiply $4$ by $2$ .

Step 14.1.4.2

Multiply $- 1$ by $8$ .

Step 14.2

Subtract $8$ from $8$ .

Step 14.3

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

Step 14.4

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

Step 14.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 15

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 15.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 2 \sqrt{2}) \cup (- 2 \sqrt{2}, 0) \cup (0, 2 \sqrt{2}) \cup (2 \sqrt{2}, \infty)$

Step 15.2

Substitute any number, such as $- 5$ , from the interval $(- \infty, - 2 \sqrt{2})$ in the first derivative $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$ to check if the result is negative or positive.

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Step 15.2.1

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = - \frac{2 (- 5) (8 - {(- 5)}^{2})}{| 8 - {(- 5)}^{2} |}$

Step 15.2.2

Simplify the result.

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Step 15.2.2.1

Multiply $2$ by $- 5$ .

$f' (- 5) = - \frac{- 10 (8 - {(- 5)}^{2})}{| 8 - {(- 5)}^{2} |}$

Step 15.2.2.2

Simplify the denominator.

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Step 15.2.2.2.1

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

$f' (- 5) = - \frac{- 10 (8 - {(- 5)}^{2})}{| 8 - 1 \cdot 25 |}$

Step 15.2.2.2.2

Multiply $- 1$ by $25$ .

$f' (- 5) = - \frac{- 10 (8 - {(- 5)}^{2})}{| 8 - 25 |}$

Step 15.2.2.2.3

Subtract $25$ from $8$ .

$f' (- 5) = - \frac{- 10 (8 - {(- 5)}^{2})}{| - 17 |}$

Step 15.2.2.2.4

The absolute value is the distance between a number and zero. The distance between $- 17$ and $0$ is $17$ .

$f' (- 5) = - \frac{- 10 (8 - {(- 5)}^{2})}{17}$

Step 15.2.2.3

Simplify the numerator.

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Step 15.2.2.3.1

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

$f' (- 5) = - \frac{- 10 (8 - 1 \cdot 25)}{17}$

Step 15.2.2.3.2

Multiply $- 1$ by $25$ .

$f' (- 5) = - \frac{- 10 (8 - 25)}{17}$

Step 15.2.2.3.3

Subtract $25$ from $8$ .

$f' (- 5) = - \frac{- 10 \cdot - 17}{17}$

Step 15.2.2.4

Simplify the expression.

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Step 15.2.2.4.1

Multiply $- 10$ by $- 17$ .

$f' (- 5) = - \frac{170}{17}$

Step 15.2.2.4.2

Divide $170$ by $17$ .

$f' (- 5) = - 1 \cdot 10$

Step 15.2.2.4.3

Multiply $- 1$ by $10$ .

$f' (- 5) = - 10$

Step 15.2.2.5

The final answer is $- 10$ .

$- 10$

Step 15.3

Substitute any number, such as $- 1$ , from the interval $(- 2 \sqrt{2}, 0)$ in the first derivative $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$ to check if the result is negative or positive.

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Step 15.3.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{2 (- 1) (8 - {(- 1)}^{2})}{| 8 - {(- 1)}^{2} |}$

Step 15.3.2

Simplify the result.

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Step 15.3.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 15.3.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 15.3.2.1.1.1

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

$f' (- 1) = - \frac{2 (- 1) (8 + (- 1) {(- 1)}^{2})}{| 8 - {(- 1)}^{2} |}$

Step 15.3.2.1.1.2

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

$f' (- 1) = - \frac{2 (- 1) (8 + {(- 1)}^{1 + 2})}{| 8 - {(- 1)}^{2} |}$

Step 15.3.2.1.2

Add $1$ and $2$ .

$f' (- 1) = - \frac{2 (- 1) (8 + {(- 1)}^{3})}{| 8 - {(- 1)}^{2} |}$

Step 15.3.2.2

Multiply $2$ by $- 1$ .

$f' (- 1) = - \frac{- 2 (8 + {(- 1)}^{3})}{| 8 - {(- 1)}^{2} |}$

Step 15.3.2.3

Simplify the denominator.

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Step 15.3.2.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 15.3.2.3.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 15.3.2.3.1.1.1

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

$f' (- 1) = - \frac{- 2 (8 + {(- 1)}^{3})}{| 8 + (- 1) {(- 1)}^{2} |}$

Step 15.3.2.3.1.1.2

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

$f' (- 1) = - \frac{- 2 (8 + {(- 1)}^{3})}{| 8 + {(- 1)}^{1 + 2} |}$

Step 15.3.2.3.1.2

Add $1$ and $2$ .

$f' (- 1) = - \frac{- 2 (8 + {(- 1)}^{3})}{| 8 + {(- 1)}^{3} |}$

Step 15.3.2.3.2

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

$f' (- 1) = - \frac{- 2 (8 + {(- 1)}^{3})}{| 8 - 1 |}$

Step 15.3.2.3.3

Subtract $1$ from $8$ .

$f' (- 1) = - \frac{- 2 (8 + {(- 1)}^{3})}{| 7 |}$

Step 15.3.2.3.4

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$f' (- 1) = - \frac{- 2 (8 + {(- 1)}^{3})}{7}$

Step 15.3.2.4

Simplify the numerator.

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Step 15.3.2.4.1

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

$f' (- 1) = - \frac{- 2 (8 - 1)}{7}$

Step 15.3.2.4.2

Subtract $1$ from $8$ .

$f' (- 1) = - \frac{- 2 \cdot 7}{7}$

Step 15.3.2.5

Simplify the expression.

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Step 15.3.2.5.1

Multiply $- 2$ by $7$ .

$f' (- 1) = - \frac{- 14}{7}$

Step 15.3.2.5.2

Divide $- 14$ by $7$ .

$f' (- 1) = 2$

Step 15.3.2.5.3

Multiply $- 1$ by $- 2$ .

$f' (- 1) = 2$

Step 15.3.2.6

The final answer is $2$ .

$2$

Step 15.4

Substitute any number, such as $1$ , from the interval $(0, 2 \sqrt{2})$ in the first derivative $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$ to check if the result is negative or positive.

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Step 15.4.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = - \frac{2 (1) (8 - {(1)}^{2})}{| 8 - {(1)}^{2} |}$

Step 15.4.2

Simplify the result.

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Step 15.4.2.1

Multiply $2$ by $1$ .

$f' (1) = - \frac{2 (8 - 1^{2})}{| 8 - 1^{2} |}$

Step 15.4.2.2

Simplify the denominator.

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Step 15.4.2.2.1

One to any power is one.

$f' (1) = - \frac{2 (8 - 1^{2})}{| 8 - 1 \cdot 1 |}$

Step 15.4.2.2.2

Multiply $- 1$ by $1$ .

$f' (1) = - \frac{2 (8 - 1^{2})}{| 8 - 1 |}$

Step 15.4.2.2.3

Subtract $1$ from $8$ .

$f' (1) = - \frac{2 (8 - 1^{2})}{| 7 |}$

Step 15.4.2.2.4

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$f' (1) = - \frac{2 (8 - 1^{2})}{7}$

Step 15.4.2.3

Simplify the numerator.

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Step 15.4.2.3.1

One to any power is one.

$f' (1) = - \frac{2 (8 - 1 \cdot 1)}{7}$

Step 15.4.2.3.2

Multiply $- 1$ by $1$ .

$f' (1) = - \frac{2 (8 - 1)}{7}$

Step 15.4.2.3.3

Subtract $1$ from $8$ .

$f' (1) = - \frac{2 \cdot 7}{7}$

Step 15.4.2.4

Simplify the expression.

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Step 15.4.2.4.1

Multiply $2$ by $7$ .

$f' (1) = - \frac{14}{7}$

Step 15.4.2.4.2

Divide $14$ by $7$ .

$f' (1) = - 1 \cdot 2$

Step 15.4.2.4.3

Multiply $- 1$ by $2$ .

$f' (1) = - 2$

Step 15.4.2.5

The final answer is $- 2$ .

$- 2$

Step 15.5

Substitute any number, such as $5$ , from the interval $(2 \sqrt{2}, \infty)$ in the first derivative $- \frac{2 x (8 - x^{2})}{| 8 - x^{2} |}$ to check if the result is negative or positive.

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Step 15.5.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = - \frac{2 (5) (8 - {(5)}^{2})}{| 8 - {(5)}^{2} |}$

Step 15.5.2

Simplify the result.

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Step 15.5.2.1

Multiply $2$ by $5$ .

$f' (5) = - \frac{10 (8 - 5^{2})}{| 8 - 5^{2} |}$

Step 15.5.2.2

Simplify the denominator.

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Step 15.5.2.2.1

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

$f' (5) = - \frac{10 (8 - 5^{2})}{| 8 - 1 \cdot 25 |}$

Step 15.5.2.2.2

Multiply $- 1$ by $25$ .

$f' (5) = - \frac{10 (8 - 5^{2})}{| 8 - 25 |}$

Step 15.5.2.2.3

Subtract $25$ from $8$ .

$f' (5) = - \frac{10 (8 - 5^{2})}{| - 17 |}$

Step 15.5.2.2.4

The absolute value is the distance between a number and zero. The distance between $- 17$ and $0$ is $17$ .

$f' (5) = - \frac{10 (8 - 5^{2})}{17}$

Step 15.5.2.3

Simplify the numerator.

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Step 15.5.2.3.1

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

$f' (5) = - \frac{10 (8 - 1 \cdot 25)}{17}$

Step 15.5.2.3.2

Multiply $- 1$ by $25$ .

$f' (5) = - \frac{10 (8 - 25)}{17}$

Step 15.5.2.3.3

Subtract $25$ from $8$ .

$f' (5) = - \frac{10 \cdot - 17}{17}$

Step 15.5.2.4

Simplify the expression.

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Step 15.5.2.4.1

Multiply $10$ by $- 17$ .

$f' (5) = - \frac{- 170}{17}$

Step 15.5.2.4.2

Divide $- 170$ by $17$ .

$f' (5) = 10$

Step 15.5.2.4.3

Multiply $- 1$ by $- 10$ .

$f' (5) = 10$

Step 15.5.2.5

The final answer is $10$ .

$10$

Step 15.6

Since the first derivative changed signs from negative to positive around $x = - 2 \sqrt{2}$ , then $x = - 2 \sqrt{2}$ is a local minimum.

$x = - 2 \sqrt{2}$ is a local minimum

Step 15.7

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 15.8

Since the first derivative changed signs from negative to positive around $x = 2 \sqrt{2}$ , then $x = 2 \sqrt{2}$ is a local minimum.

$x = 2 \sqrt{2}$ is a local minimum

Step 15.9

These are the local extrema for $f (x) = | 8 - x^{2} |$ .

$x = - 2 \sqrt{2}$ is a local minimum

$x = 0$ is a local maximum

$x = 2 \sqrt{2}$ is a local minimum

$x = - 2 \sqrt{2}$ is a local minimum

$x = 0$ is a local maximum

$x = 2 \sqrt{2}$ is a local minimum

Step 16