Calculus Examples

$y = 2 | x^{2} - 16 |$

Step 1

Write $y = 2 | x^{2} - 16 |$ as a function.

$f (x) = 2 | x^{2} - 16 |$

Step 2

Find the first derivative of the function.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 | x^{2} - 16 |$ with respect to $x$ is $2 \frac{d}{d x} [| x^{2} - 16 |]$ .

$2 \frac{d}{d x} [| x^{2} - 16 |]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 16$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 16$ .

$2 (\frac{d}{d u} [| u |] \frac{d}{d x} [x^{2} - 16])$

Step 2.2.2

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

$2 (\frac{u}{| u |} \frac{d}{d x} [x^{2} - 16])$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} - 16$ .

$2 (\frac{x^{2} - 16}{| x^{2} - 16 |} \frac{d}{d x} [x^{2} - 16])$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\frac{x^{2} - 16}{| x^{2} - 16 |}$ and $2$ .

$\frac{(x^{2} - 16) \cdot 2}{| x^{2} - 16 |} \frac{d}{d x} [x^{2} - 16]$

Step 2.3.2

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

$\frac{2 \cdot (x^{2} - 16)}{| x^{2} - 16 |} \frac{d}{d x} [x^{2} - 16]$

Step 2.3.3

By the Sum Rule, the derivative of $x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]$ .

$\frac{2 (x^{2} - 16)}{| x^{2} - 16 |} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16])$

Step 2.3.4

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

$\frac{2 (x^{2} - 16)}{| x^{2} - 16 |} (2 x + \frac{d}{d x} [- 16])$

Step 2.3.5

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$\frac{2 (x^{2} - 16)}{| x^{2} - 16 |} (2 x + 0)$

Step 2.3.6

Combine fractions.

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Step 2.3.6.1

Add $2 x$ and $0$ .

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

Step 2.3.6.2

Combine $2$ and $\frac{2 (x^{2} - 16)}{| x^{2} - 16 |}$ .

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

Step 2.3.6.3

Multiply $2$ by $2$ .

$\frac{4 (x^{2} - 16)}{| x^{2} - 16 |} x$

Step 2.3.6.4

Combine $\frac{4 (x^{2} - 16)}{| x^{2} - 16 |}$ and $x$ .

$\frac{4 (x^{2} - 16) x}{| x^{2} - 16 |}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$\frac{(4 x^{2} + 4 \cdot - 16) x}{| x^{2} - 16 |}$

Step 2.4.2

Apply the distributive property.

$\frac{4 x^{2} x + 4 \cdot - 16 x}{| x^{2} - 16 |}$

Step 2.4.3

Simplify each term.

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Step 2.4.3.1

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

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Step 2.4.3.1.1

Move $x$ .

$\frac{4 (x \cdot x^{2}) + 4 \cdot - 16 x}{| x^{2} - 16 |}$

Step 2.4.3.1.2

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

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Step 2.4.3.1.2.1

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

$\frac{4 (x^{1} x^{2}) + 4 \cdot - 16 x}{| x^{2} - 16 |}$

Step 2.4.3.1.2.2

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

$\frac{4 x^{1 + 2} + 4 \cdot - 16 x}{| x^{2} - 16 |}$

Step 2.4.3.1.3

Add $1$ and $2$ .

$\frac{4 x^{3} + 4 \cdot - 16 x}{| x^{2} - 16 |}$

Step 2.4.3.2

Multiply $4$ by $- 16$ .

$\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$

Step 3

Find the second derivative of the function.

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Step 3.1

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 $f (x) = 4 x^{3} - 64 x$ and $g (x) = | x^{2} - 16 |$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $4 x^{3} - 64 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 64 x]$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $3$ by $4$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} + \frac{d}{d x} (- 64 x)) - (4 x^{3} - 64 x) \frac{d}{d x} | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.2.5

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64 x$ with respect to $x$ is $- 64 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64 \frac{d}{d x} x) - (4 x^{3} - 64 x) \frac{d}{d x} | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.2.6

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

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64 \cdot 1) - (4 x^{3} - 64 x) \frac{d}{d x} | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.2.7

Multiply $- 64$ by $1$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) \frac{d}{d x} | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 16$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 16$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{d}{d u} (| u |) \frac{d}{d x} (x^{2} - 16))}{{| x^{2} - 16 |}^{2}}$

Step 3.3.2

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

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{u}{| u |} \cdot \frac{d}{d x} (x^{2} - 16))}{{| x^{2} - 16 |}^{2}}$

Step 3.3.3

Replace all occurrences of $u$ with $x^{2} - 16$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{x^{2} - 16}{| x^{2} - 16 |} \cdot \frac{d}{d x} (x^{2} - 16))}{{| x^{2} - 16 |}^{2}}$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{x^{2} - 16}{| x^{2} - 16 |} \cdot (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 16)))}{{| x^{2} - 16 |}^{2}}$

Step 3.4.2

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

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{x^{2} - 16}{| x^{2} - 16 |} \cdot (2 x + \frac{d}{d x} (- 16)))}{{| x^{2} - 16 |}^{2}}$

Step 3.4.3

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{x^{2} - 16}{| x^{2} - 16 |} \cdot (2 x + 0))}{{| x^{2} - 16 |}^{2}}$

Step 3.4.4

Combine fractions.

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Step 3.4.4.1

Add $2 x$ and $0$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{x^{2} - 16}{| x^{2} - 16 |} \cdot (2 x))}{{| x^{2} - 16 |}^{2}}$

Step 3.4.4.2

Combine $2$ and $\frac{x^{2} - 16}{| x^{2} - 16 |}$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) (\frac{2 (x^{2} - 16)}{| x^{2} - 16 |} \cdot x)}{{| x^{2} - 16 |}^{2}}$

Step 3.4.4.3

Combine $\frac{2 (x^{2} - 16)}{| x^{2} - 16 |}$ and $x$ .

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) - (4 x^{3} - 64 x) \frac{2 (x^{2} - 16) x}{| x^{2} - 16 |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) + (- (4 x^{3}) - (- 64 x)) (\frac{2 (x^{2} - 16) x}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.2

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) + (- (4 x^{3}) - (- 64 x)) (\frac{(2 x^{2} + 2 \cdot - 16) x}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.3

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2} - 64) + (- (4 x^{3}) - (- 64 x)) (\frac{2 x^{2} x + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4

Simplify the numerator.

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Step 3.5.4.1

Simplify each term.

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Step 3.5.4.1.1

Apply the distributive property.

$f'' (x) = \frac{| x^{2} - 16 | (12 x^{2}) + | x^{2} - 16 | \cdot - 64 + (- (4 x^{3}) - (- 64 x)) (\frac{2 x^{2} x + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} + | x^{2} - 16 | \cdot - 64 + (- (4 x^{3}) - (- 64 x)) (\frac{2 x^{2} x + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.3

Move $- 64$ to the left of $| x^{2} - 16 |$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 \cdot | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x^{2} x + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4

Simplify the numerator.

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Step 3.5.4.1.4.1

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

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Step 3.5.4.1.4.1.1

Move $x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 (x \cdot x^{2}) + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.1.2

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

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Step 3.5.4.1.4.1.2.1

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 (x \cdot x^{2}) + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.1.2.2

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x^{1 + 2} + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x^{3} + 2 \cdot (- 16 x)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.2

Multiply $2$ by $- 16$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x^{3} - 32 x}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.3

Rewrite $2 x^{3} - 32 x$ in a factored form.

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Step 3.5.4.1.4.3.1

Factor $2 x$ out of $2 x^{3} - 32 x$ .

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Step 3.5.4.1.4.3.1.1

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x^{2}) - 32 x}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.3.1.2

Factor $2 x$ out of $- 32 x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x^{2}) + 2 x (- 16)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.3.1.3

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 16)$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x^{2} - 16)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.3.2

Rewrite $16$ as $4^{2}$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x^{2} - 4^{2})}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.4.3.3

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 $b = 4$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5

Simplify the denominator.

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Step 3.5.4.1.5.1

Rewrite $16$ as $4^{2}$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x^{2} - 4^{2} |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.2

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 $b = 4$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| (x + 4) (x - 4) |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.3

Expand $(x + 4) (x - 4)$ using the FOIL Method.

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Step 3.5.4.1.5.3.1

Apply the distributive property.

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x (x - 4) + 4 (x - 4) |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.3.2

Apply the distributive property.

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x \cdot x + x \cdot - 4 + 4 (x - 4) |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.3.3

Apply the distributive property.

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.4

Combine the opposite terms in $x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4$ .

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Step 3.5.4.1.5.4.1

Reorder the factors in the terms $x \cdot - 4$ and $4 x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x \cdot x - 4 x + 4 x + 4 \cdot - 4 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.4.2

Add $- 4 x$ and $4 x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x \cdot x + 0 + 4 \cdot - 4 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.4.3

Add $x \cdot x$ and $0$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x \cdot x + 4 \cdot - 4 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.5

Simplify each term.

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Step 3.5.4.1.5.5.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x^{2} + 4 \cdot - 4 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.5.2

Multiply $4$ by $- 4$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x^{2} - 16 |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.6

Rewrite $16$ as $4^{2}$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| x^{2} - 4^{2} |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.5.7

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 $b = 4$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- (4 x^{3}) - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| (x + 4) (x - 4) |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.6

Simplify each term.

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Step 3.5.4.1.6.1

Multiply $4$ by $- 1$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- 4 x^{3} - (- 64 x)) (\frac{2 x (x + 4) (x - 4)}{| (x + 4) (x - 4) |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.6.2

Multiply $- 64$ by $- 1$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + (- 4 x^{3} + 64 x) (\frac{2 x (x + 4) (x - 4)}{| (x + 4) (x - 4) |})}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.7

Multiply $- 4 x^{3} + 64 x$ by $\frac{2 x (x + 4) (x - 4)}{| (x + 4) (x - 4) |}$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{(- 4 x^{3} + 64 x) (2 x (x + 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8

Simplify the numerator.

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Step 3.5.4.1.8.1

Factor $4 x$ out of $- 4 x^{3} + 64 x$ .

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Step 3.5.4.1.8.1.1

Factor $4 x$ out of $- 4 x^{3}$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{(4 x (- x^{2}) + 64 x) \cdot (2 x (x + 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.1.2

Factor $4 x$ out of $64 x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{(4 x (- x^{2}) + 4 x (16)) \cdot (2 x (x + 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.1.3

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{4 x (- x^{2} + 16) \cdot (2 x (x + 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.2

Rewrite $16$ as $4^{2}$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{4 x (- x^{2} + 4^{2}) \cdot (2 x (x + 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.3

Reorder $- x^{2}$ and $4^{2}$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{4 x (4^{2} - x^{2}) \cdot (2 x (x + 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{4 x (4 + x) (4 - x) \cdot (2 x (x + 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5

Combine exponents.

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Step 3.5.4.1.8.5.1

Multiply $2$ by $4$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x (4 + x) (4 - x) x (x + 4) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.2

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 (x \cdot x) (4 + x) (4 - x) (x + 4) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.3

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 (x \cdot x) (4 + x) (4 - x) (x + 4) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.4

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{1 + 1} (4 + x) (4 - x) (x + 4) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.5

Add $1$ and $1$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} (4 + x) (4 - x) (x + 4) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.6

Reorder terms.

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} (x + 4) (4 - x) (x + 4) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.7

Raise $x + 4$ to the power of $1$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} ((x + 4) (x + 4)) (4 - x) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.8

Raise $x + 4$ to the power of $1$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} ((x + 4) (x + 4)) (4 - x) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.9

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{1 + 1} (4 - x) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.10

Add $1$ and $1$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} (4 - x) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.11

Rewrite $4$ as $- 1 (- 4)$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} (- 1 \cdot - 4 - x) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.12

Factor $- 1$ out of $- x$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} (- 1 \cdot - 4 - (x)) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.13

Factor $- 1$ out of $- 1 (- 4) - (x)$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} (- 1 (- 4 + x)) (x - 4)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.14

Reorder terms.

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} \cdot (- 1 (x - 4) (x - 4))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.15

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} \cdot (- 1 ((x - 4) (x - 4)))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.16

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} \cdot (- 1 ((x - 4) (x - 4)))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.17

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

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} \cdot (- 1 {(x - 4)}^{1 + 1})}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.18

Add $1$ and $1$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{8 x^{2} {(x + 4)}^{2} \cdot (- 1 {(x - 4)}^{2})}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.8.5.19

Multiply $- 1$ by $8$ .

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | + \frac{- 8 x^{2} {(x + 4)}^{2} {(x - 4)}^{2}}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.1.9

Move the negative in front of the fraction.

$f'' (x) = \frac{12 | x^{2} - 16 | x^{2} - 64 | x^{2} - 16 | - \frac{8 x^{2} {(x + 4)}^{2} {(x - 4)}^{2}}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.2

To write $12 | x^{2} - 16 | x^{2}$ as a fraction with a common denominator, multiply by $\frac{| (x + 4) (x - 4) |}{| (x + 4) (x - 4) |}$ .

$f'' (x) = \frac{\frac{12 | x^{2} - 16 | x^{2} | (x + 4) (x - 4) |}{| (x + 4) (x - 4) |} - \frac{8 x^{2} {(x + 4)}^{2} {(x - 4)}^{2}}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{12 | x^{2} - 16 | x^{2} | (x + 4) (x - 4) | - 8 x^{2} {(x + 4)}^{2} {(x - 4)}^{2}}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4

Simplify each term.

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Step 3.5.4.4.1

Simplify the numerator.

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Step 3.5.4.4.1.1

Factor $4 x^{2}$ out of $12 | x^{2} - 16 | x^{2} | (x + 4) (x - 4) | - 8 x^{2} {(x + 4)}^{2} {(x - 4)}^{2}$ .

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Step 3.5.4.4.1.1.1

Factor $4 x^{2}$ out of $12 | x^{2} - 16 | x^{2} | (x + 4) (x - 4) |$ .

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

Step 3.5.4.4.1.1.2

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

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

Step 3.5.4.4.1.1.3

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

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

Step 3.5.4.4.1.2

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

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

Step 3.5.4.4.1.3

Simplify each term.

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Step 3.5.4.4.1.3.1

Expand $(x + 4) (x - 4)$ using the FOIL Method.

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Step 3.5.4.4.1.3.1.1

Apply the distributive property.

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

Step 3.5.4.4.1.3.1.2

Apply the distributive property.

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

Step 3.5.4.4.1.3.1.3

Apply the distributive property.

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

Step 3.5.4.4.1.3.2

Combine the opposite terms in $x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4$ .

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Step 3.5.4.4.1.3.2.1

Reorder the factors in the terms $x \cdot - 4$ and $4 x$ .

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

Step 3.5.4.4.1.3.2.2

Add $- 4 x$ and $4 x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | (x \cdot x + 0 + 4 \cdot - 4) (x^{2} - 16) | - 2 {(x + 4)}^{2} {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.2.3

Add $x \cdot x$ and $0$ .

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

Step 3.5.4.4.1.3.3

Simplify each term.

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Step 3.5.4.4.1.3.3.1

Multiply $x$ by $x$ .

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

Step 3.5.4.4.1.3.3.2

Multiply $4$ by $- 4$ .

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

Step 3.5.4.4.1.3.4

Expand $(x^{2} - 16) (x^{2} - 16)$ using the FOIL Method.

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Step 3.5.4.4.1.3.4.1

Apply the distributive property.

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

Step 3.5.4.4.1.3.4.2

Apply the distributive property.

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

Step 3.5.4.4.1.3.4.3

Apply the distributive property.

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

Step 3.5.4.4.1.3.5

Simplify and combine like terms.

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Step 3.5.4.4.1.3.5.1

Simplify each term.

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Step 3.5.4.4.1.3.5.1.1

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

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Step 3.5.4.4.1.3.5.1.1.1

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

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

Step 3.5.4.4.1.3.5.1.1.2

Add $2$ and $2$ .

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

Step 3.5.4.4.1.3.5.1.2

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

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

Step 3.5.4.4.1.3.5.1.3

Multiply $- 16$ by $- 16$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 16 x^{2} - 16 x^{2} + 256 | - 2 {(x + 4)}^{2} {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.5.2

Subtract $16 x^{2}$ from $- 16 x^{2}$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 {(x + 4)}^{2} {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.6

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

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 ((x + 4) (x + 4)) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.7

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 3.5.4.4.1.3.7.1

Apply the distributive property.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x (x + 4) + 4 (x + 4)) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.7.2

Apply the distributive property.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x \cdot x + x \cdot 4 + 4 (x + 4)) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.7.3

Apply the distributive property.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.8

Simplify and combine like terms.

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Step 3.5.4.4.1.3.8.1

Simplify each term.

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Step 3.5.4.4.1.3.8.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.8.1.2

Move $4$ to the left of $x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.8.1.3

Multiply $4$ by $4$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x^{2} + 4 x + 4 x + 16) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.8.2

Add $4 x$ and $4 x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x^{2} + 8 x + 16) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.9

Apply the distributive property.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 2 (8 x) - 2 \cdot 16) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.10

Simplify.

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Step 3.5.4.4.1.3.10.1

Multiply $8$ by $- 2$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 2 \cdot 16) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.10.2

Multiply $- 2$ by $16$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) {(x - 4)}^{2})}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.11

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

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) ((x - 4) (x - 4)))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.12

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 3.5.4.4.1.3.12.1

Apply the distributive property.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) (x (x - 4) - 4 (x - 4)))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.12.2

Apply the distributive property.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) (x \cdot x + x \cdot - 4 - 4 (x - 4)))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.12.3

Apply the distributive property.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.13

Simplify and combine like terms.

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Step 3.5.4.4.1.3.13.1

Simplify each term.

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Step 3.5.4.4.1.3.13.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.13.1.2

Move $- 4$ to the left of $x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.13.1.3

Multiply $- 4$ by $- 4$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) (x^{2} - 4 x - 4 x + 16))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.13.2

Subtract $4 x$ from $- 4 x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | + (- 2 x^{2} - 16 x - 32) (x^{2} - 8 x + 16))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.14

Expand $(- 2 x^{2} - 16 x - 32) (x^{2} - 8 x + 16)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{2} x^{2} - 2 x^{2} (- 8 x) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15

Simplify each term.

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Step 3.5.4.4.1.3.15.1

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

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Step 3.5.4.4.1.3.15.1.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 (x^{2} x^{2}) - 2 x^{2} (- 8 x) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.1.2

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

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{2 + 2} - 2 x^{2} (- 8 x) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.1.3

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 2 x^{2} (- 8 x) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 2 \cdot (- 8 x^{2} x) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.3

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

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Step 3.5.4.4.1.3.15.3.1

Move $x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 2 \cdot (- 8 (x \cdot x^{2})) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.3.2

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

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Step 3.5.4.4.1.3.15.3.2.1

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

Step 3.5.4.4.1.3.15.3.2.2

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

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 2 \cdot (- 8 x^{1 + 2}) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 2 \cdot (- 8 x^{3}) - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.4

Multiply $- 2$ by $- 8$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 2 x^{2} \cdot 16 - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.5

Multiply $16$ by $- 2$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x \cdot x^{2} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.6

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

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Step 3.5.4.4.1.3.15.6.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 (x^{2} x) - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.6.2

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

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Step 3.5.4.4.1.3.15.6.2.1

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

Step 3.5.4.4.1.3.15.6.2.2

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

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{2 + 1} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.6.3

Add $2$ and $1$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} - 16 x (- 8 x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} - 16 \cdot (- 8 x \cdot x) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.8

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

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Step 3.5.4.4.1.3.15.8.1

Move $x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} - 16 \cdot (- 8 (x \cdot x)) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.8.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} - 16 \cdot (- 8 x^{2}) - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.9

Multiply $- 16$ by $- 8$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} + 128 x^{2} - 16 x \cdot 16 - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.10

Multiply $16$ by $- 16$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} + 128 x^{2} - 256 x - 32 x^{2} - 32 (- 8 x) - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.11

Multiply $- 8$ by $- 32$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} + 128 x^{2} - 256 x - 32 x^{2} + 256 x - 32 \cdot 16)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.15.12

Multiply $- 32$ by $16$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} + 128 x^{2} - 256 x - 32 x^{2} + 256 x - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.16

Combine the opposite terms in $- 2 x^{4} + 16 x^{3} - 32 x^{2} - 16 x^{3} + 128 x^{2} - 256 x - 32 x^{2} + 256 x - 512$ .

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Step 3.5.4.4.1.3.16.1

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

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 32 x^{2} + 0 + 128 x^{2} - 256 x - 32 x^{2} + 256 x - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.16.2

Add $- 2 x^{4} - 32 x^{2}$ and $0$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 32 x^{2} + 128 x^{2} - 256 x - 32 x^{2} + 256 x - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.16.3

Add $- 256 x$ and $256 x$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 32 x^{2} + 128 x^{2} - 32 x^{2} + 0 - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.16.4

Add $- 2 x^{4} - 32 x^{2} + 128 x^{2} - 32 x^{2}$ and $0$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} - 32 x^{2} + 128 x^{2} - 32 x^{2} - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.17

Add $- 32 x^{2}$ and $128 x^{2}$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 96 x^{2} - 32 x^{2} - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.3.18

Subtract $32 x^{2}$ from $96 x^{2}$ .

$f'' (x) = \frac{\frac{4 x^{2} (3 | x^{4} - 32 x^{2} + 256 | - 2 x^{4} + 64 x^{2} - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.4

Reorder terms.

$f'' (x) = \frac{\frac{4 x^{2} (- 2 x^{4} + 64 x^{2} + 3 | x^{4} - 32 x^{2} + 256 | - 512)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5

Rewrite $- 2 x^{4} + 64 x^{2} + 3 | x^{4} - 32 x^{2} + 256 | - 512$ in a factored form.

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Step 3.5.4.4.1.5.1

Regroup terms.

$f'' (x) = \frac{\frac{4 x^{2} (- 512 - 2 x^{4} + 64 x^{2} + 3 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.2

Factor $- 1$ out of $- 2 x^{4} + 64 x^{2} + 3 | x^{4} - 32 x^{2} + 256 |$ .

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Step 3.5.4.4.1.5.2.1

Factor $- 1$ out of $- 2 x^{4}$ .

$f'' (x) = \frac{\frac{4 x^{2} (- 512 - (2 x^{4}) + 64 x^{2} + 3 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.2.2

Factor $- 1$ out of $64 x^{2}$ .

$f'' (x) = \frac{\frac{4 x^{2} (- 512 - (2 x^{4}) - (- 64 x^{2}) + 3 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.2.3

Factor $- 1$ out of $3 | x^{4} - 32 x^{2} + 256 |$ .

$f'' (x) = \frac{\frac{4 x^{2} (- 512 - (2 x^{4}) - (- 64 x^{2}) - (- 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.2.4

Factor $- 1$ out of $- (2 x^{4}) - (- 64 x^{2})$ .

$f'' (x) = \frac{\frac{4 x^{2} (- 512 - (2 x^{4} - 64 x^{2}) - (- 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.2.5

Factor $- 1$ out of $- (2 x^{4} - 64 x^{2}) - (- 3 | x^{4} - 32 x^{2} + 256 |)$ .

$f'' (x) = \frac{\frac{4 x^{2} (- 512 - (2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.3

Factor $- 1$ out of $- 512 - (2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |)$ .

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Step 3.5.4.4.1.5.3.1

Rewrite $- 512$ as $- 1 (512)$ .

$f'' (x) = \frac{\frac{4 x^{2} (- 1 \cdot 512 - (2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.3.2

Factor $- 1$ out of $- 1 (512) - (2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |)$ .

$f'' (x) = \frac{\frac{4 x^{2} (- 1 (512 + 2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.3.3

Rewrite $- 1 (512 + 2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |)$ as $- (512 + 2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |)$ .

$f'' (x) = \frac{\frac{4 x^{2} (- (512 + 2 x^{4} - 64 x^{2} - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.5.4

Reorder terms.

$f'' (x) = \frac{\frac{4 x^{2} (- (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

$f'' (x) = \frac{\frac{4 x^{2} \cdot (- 1 (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.6

Combine exponents.

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Step 3.5.4.4.1.6.1

Factor out negative.

$f'' (x) = \frac{\frac{- (4 x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.1.6.2

Multiply $4$ by $- 1$ .

$f'' (x) = \frac{\frac{- 4 (x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.4.2

Move the negative in front of the fraction.

$f'' (x) = \frac{- \frac{4 x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 |}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.5

To write $- 64 | x^{2} - 16 |$ as a fraction with a common denominator, multiply by $\frac{| (x + 4) (x - 4) |}{| (x + 4) (x - 4) |}$ .

$f'' (x) = \frac{- \frac{4 x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |} - 64 | x^{2} - 16 | \cdot \frac{| (x + 4) (x - 4) |}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.6

Combine $- 64 | x^{2} - 16 |$ and $\frac{| (x + 4) (x - 4) |}{| (x + 4) (x - 4) |}$ .

$f'' (x) = \frac{- \frac{4 x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |} + \frac{- 64 | x^{2} - 16 | | (x + 4) (x - 4) |}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.7

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{- 4 x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |) - 64 | x^{2} - 16 | | (x + 4) (x - 4) |}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8

Simplify the numerator.

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Step 3.5.4.8.1

Factor $4$ out of $- 4 x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |) - 64 | x^{2} - 16 | | (x + 4) (x - 4) |$ .

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Step 3.5.4.8.1.1

Factor $4$ out of $- 4 x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |)$ .

$f'' (x) = \frac{\frac{4 (- x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |)) - 64 | x^{2} - 16 | | (x + 4) (x - 4) |}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.1.2

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

$f'' (x) = \frac{\frac{4 (- x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |)) + 4 (- 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.1.3

Factor $4$ out of $4 (- x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |)) + 4 (- 16 | x^{2} - 16 | | (x + 4) (x - 4) |)$ .

$f'' (x) = \frac{\frac{4 (- x^{2} (2 x^{4} - 64 x^{2} + 512 - 3 | x^{4} - 32 x^{2} + 256 |) - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.2

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (- x^{2} (2 x^{4}) - x^{2} (- 64 x^{2}) - x^{2} \cdot 512 - x^{2} (- 3 | x^{4} - 32 x^{2} + 256 |) - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.3

Simplify.

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Step 3.5.4.8.3.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{4 (- 1 \cdot (2 x^{2} x^{4}) - x^{2} (- 64 x^{2}) - x^{2} \cdot 512 - x^{2} (- 3 | x^{4} - 32 x^{2} + 256 |) - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.3.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{4 (- 1 \cdot (2 x^{2} x^{4}) - 1 \cdot (- 64 x^{2} x^{2}) - x^{2} \cdot 512 - x^{2} (- 3 | x^{4} - 32 x^{2} + 256 |) - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.3.3

Multiply $512$ by $- 1$ .

$f'' (x) = \frac{\frac{4 (- 1 \cdot (2 x^{2} x^{4}) - 1 \cdot (- 64 x^{2} x^{2}) - 512 x^{2} - x^{2} (- 3 | x^{4} - 32 x^{2} + 256 |) - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.3.4

Multiply $- 3$ by $- 1$ .

$f'' (x) = \frac{\frac{4 (- 1 \cdot (2 x^{2} x^{4}) - 1 \cdot (- 64 x^{2} x^{2}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4

Simplify each term.

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Step 3.5.4.8.4.1

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

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Step 3.5.4.8.4.1.1

Move $x^{4}$ .

$f'' (x) = \frac{\frac{4 (- 1 \cdot (2 (x^{4} x^{2})) - 1 \cdot (- 64 x^{2} x^{2}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4.1.2

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

$f'' (x) = \frac{\frac{4 (- 1 \cdot (2 x^{4 + 2}) - 1 \cdot (- 64 x^{2} x^{2}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4.1.3

Add $4$ and $2$ .

$f'' (x) = \frac{\frac{4 (- 1 \cdot (2 x^{6}) - 1 \cdot (- 64 x^{2} x^{2}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4.2

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} - 1 \cdot (- 64 x^{2} x^{2}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4.3

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

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Step 3.5.4.8.4.3.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} - 1 \cdot (- 64 (x^{2} x^{2})) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4.3.2

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

$f'' (x) = \frac{\frac{4 (- 2 x^{6} - 1 \cdot (- 64 x^{2 + 2}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4.3.3

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} - 1 \cdot (- 64 x^{4}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.4.4

Multiply $- 1$ by $- 64$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | (x + 4) (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.5

Expand $(x + 4) (x - 4)$ using the FOIL Method.

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Step 3.5.4.8.5.1

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x (x - 4) + 4 (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.5.2

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x \cdot x + x \cdot - 4 + 4 (x - 4) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.5.3

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.6

Combine the opposite terms in $x \cdot x + x \cdot - 4 + 4 x + 4 \cdot - 4$ .

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Step 3.5.4.8.6.1

Reorder the factors in the terms $x \cdot - 4$ and $4 x$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x \cdot x - 4 x + 4 x + 4 \cdot - 4 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.6.2

Add $- 4 x$ and $4 x$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x \cdot x + 0 + 4 \cdot - 4 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.6.3

Add $x \cdot x$ and $0$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x \cdot x + 4 \cdot - 4 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.7

Simplify each term.

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Step 3.5.4.8.7.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x^{2} + 4 \cdot - 4 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.7.2

Multiply $4$ by $- 4$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} - 16 | | x^{2} - 16 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.8

Multiply $- 16 | x^{2} - 16 | | x^{2} - 16 |$ .

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Step 3.5.4.8.8.1

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

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | (x^{2} - 16) (x^{2} - 16) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.8.2

Raise $x^{2} - 16$ to the power of $1$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | (x^{2} - 16) (x^{2} - 16) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.8.3

Raise $x^{2} - 16$ to the power of $1$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | (x^{2} - 16) (x^{2} - 16) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.8.4

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

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | {(x^{2} - 16)}^{1 + 1} |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.8.5

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | {(x^{2} - 16)}^{2} |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.9

Rewrite ${(x^{2} - 16)}^{2}$ as $(x^{2} - 16) (x^{2} - 16)$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | (x^{2} - 16) (x^{2} - 16) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.10

Expand $(x^{2} - 16) (x^{2} - 16)$ using the FOIL Method.

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Step 3.5.4.8.10.1

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} (x^{2} - 16) - 16 (x^{2} - 16) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.10.2

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} x^{2} + x^{2} \cdot - 16 - 16 (x^{2} - 16) |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.10.3

Apply the distributive property.

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2} x^{2} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.11

Simplify and combine like terms.

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Step 3.5.4.8.11.1

Simplify each term.

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Step 3.5.4.8.11.1.1

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

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Step 3.5.4.8.11.1.1.1

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

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{2 + 2} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.11.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} + x^{2} \cdot - 16 - 16 x^{2} - 16 \cdot - 16 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.11.1.2

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

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 16 \cdot x^{2} - 16 x^{2} - 16 \cdot - 16 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.11.1.3

Multiply $- 16$ by $- 16$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 16 x^{2} - 16 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.8.11.2

Subtract $16 x^{2}$ from $- 16 x^{2}$ .

$f'' (x) = \frac{\frac{4 (- 2 x^{6} + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.9

Factor $- 1$ out of $- 2 x^{6}$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6}) + 64 x^{4} - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.10

Factor $- 1$ out of $64 x^{4}$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6}) - (- 64 x^{4}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.11

Factor $- 1$ out of $- (2 x^{6}) - (- 64 x^{4})$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6} - 64 x^{4}) - 512 x^{2} + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.12

Factor $- 1$ out of $- 512 x^{2}$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6} - 64 x^{4}) - (512 x^{2}) + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.13

Factor $- 1$ out of $- (2 x^{6} - 64 x^{4}) - (512 x^{2})$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6} - 64 x^{4} + 512 x^{2}) + 3 x^{2} | x^{4} - 32 x^{2} + 256 | - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.14

Factor $- 1$ out of $3 x^{2} | x^{4} - 32 x^{2} + 256 |$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6} - 64 x^{4} + 512 x^{2}) - (- 3 x^{2} | x^{4} - 32 x^{2} + 256 |) - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.15

Factor $- 1$ out of $- (2 x^{6} - 64 x^{4} + 512 x^{2}) - (- 3 x^{2} | x^{4} - 32 x^{2} + 256 |)$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 |) - 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.16

Factor $- 1$ out of $- 16 | x^{4} - 32 x^{2} + 256 |$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 |) - (16 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.17

Factor $- 1$ out of $- (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 |) - (16 | x^{4} - 32 x^{2} + 256 |)$ .

$f'' (x) = \frac{\frac{4 (- (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.18

Rewrite $- (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |)$ as $- 1 (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |)$ .

$f'' (x) = \frac{\frac{4 (- 1 (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |))}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.4.19

Move the negative in front of the fraction.

$f'' (x) = \frac{- \frac{4 (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$

Step 3.5.5

Combine terms.

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Step 3.5.5.1

Rewrite $\frac{- \frac{4 (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}}{{| x^{2} - 16 |}^{2}}$ as a product.

$f'' (x) = - \frac{4 (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |} \cdot \frac{1}{{| x^{2} - 16 |}^{2}}$

Step 3.5.5.2

Multiply $\frac{1}{{| x^{2} - 16 |}^{2}}$ by $\frac{4 (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |)}{| (x + 4) (x - 4) |}$ .

$f'' (x) = - \frac{4 (2 x^{6} - 64 x^{4} + 512 x^{2} - 3 x^{2} | x^{4} - 32 x^{2} + 256 | + 16 | x^{4} - 32 x^{2} + 256 |)}{{| x^{2} - 16 |}^{2} | (x + 4) (x - 4) |}$

Step 4

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

$\frac{4 x^{3} - 64 x}{| x^{2} - 16 |} = 0$

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

Since $2$ is constant with respect to $x$ , the derivative of $2 | x^{2} - 16 |$ with respect to $x$ is $2 \frac{d}{d x} [| x^{2} - 16 |]$ .

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

Step 5.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 16$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 16$ .

$f' (x) = 2 (\frac{d}{d u} (| u |) \frac{d}{d x} (x^{2} - 16))$

Step 5.1.2.2

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

$f' (x) = 2 (\frac{u}{| u |} \cdot \frac{d}{d x} (x^{2} - 16))$

Step 5.1.2.3

Replace all occurrences of $u$ with $x^{2} - 16$ .

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

Step 5.1.3

Differentiate.

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Step 5.1.3.1

Combine $\frac{x^{2} - 16}{| x^{2} - 16 |}$ and $2$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

By the Sum Rule, the derivative of $x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]$ .

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

Step 5.1.3.4

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

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

Step 5.1.3.5

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 (x^{2} - 16)}{| x^{2} - 16 |} \cdot (2 x + 0)$

Step 5.1.3.6

Combine fractions.

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Step 5.1.3.6.1

Add $2 x$ and $0$ .

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

Step 5.1.3.6.2

Combine $2$ and $\frac{2 (x^{2} - 16)}{| x^{2} - 16 |}$ .

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

Step 5.1.3.6.3

Multiply $2$ by $2$ .

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

Step 5.1.3.6.4

Combine $\frac{4 (x^{2} - 16)}{| x^{2} - 16 |}$ and $x$ .

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

Step 5.1.4

Simplify.

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Step 5.1.4.1

Apply the distributive property.

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

Step 5.1.4.2

Apply the distributive property.

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

Step 5.1.4.3

Simplify each term.

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Step 5.1.4.3.1

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

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Step 5.1.4.3.1.1

Move $x$ .

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

Step 5.1.4.3.1.2

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

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Step 5.1.4.3.1.2.1

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

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

Step 5.1.4.3.1.2.2

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

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

Step 5.1.4.3.1.3

Add $1$ and $2$ .

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

Step 5.1.4.3.2

Multiply $4$ by $- 16$ .

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$ .

$\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{4 x^{3} - 64 x}{| x^{2} - 16 |} = 0$

Step 6.2

Set the numerator equal to zero.

$4 x^{3} - 64 x = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Factor the left side of the equation.

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Step 6.3.1.1

Factor $4 x$ out of $4 x^{3} - 64 x$ .

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Step 6.3.1.1.1

Factor $4 x$ out of $4 x^{3}$ .

$4 x (x^{2}) - 64 x = 0$

Step 6.3.1.1.2

Factor $4 x$ out of $- 64 x$ .

$4 x (x^{2}) + 4 x (- 16) = 0$

Step 6.3.1.1.3

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

$4 x (x^{2} - 16) = 0$

Step 6.3.1.2

Rewrite $16$ as $4^{2}$ .

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

Step 6.3.1.3

Factor.

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Step 6.3.1.3.1

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 $b = 4$ .

$4 x ((x + 4) (x - 4)) = 0$

Step 6.3.1.3.2

Remove unnecessary parentheses.

$4 x (x + 4) (x - 4) = 0$

Step 6.3.2

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

$x = 0$

$x + 4 = 0$

$x - 4 = 0$

Step 6.3.3

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 6.3.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 6.3.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 6.3.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 6.3.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 6.3.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 6.3.6

The final solution is all the values that make $4 x (x + 4) (x - 4) = 0$ true.

$x = 0, - 4, 4$

Step 6.4

Exclude the solutions that do not make $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |} = 0$ true.

$x = 0$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$ equal to $0$ to find where the expression is undefined.

$| x^{2} - 16 | = 0$

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 $| x | = \pm x$ .

$x^{2} - 16 = \pm 0$

Step 7.2.2

Plus or minus $0$ is $0$ .

$x^{2} - 16 = 0$

Step 7.2.3

Add $16$ to both sides of the equation.

$x^{2} = 16$

Step 7.2.4

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

$x = \pm \sqrt{16}$

Step 7.2.5

Simplify $\pm \sqrt{16}$ .

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Step 7.2.5.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 7.2.5.2

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

$x = \pm 4$

Step 7.2.6

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

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Step 7.2.6.1

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

$x = 4$

Step 7.2.6.2

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

$x = - 4$

Step 7.2.6.3

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

$x = 4, - 4$

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 $0$ .

$x = - 4, x = 4$

Step 8

Critical points to evaluate.

$x = 0, - 4, 4$

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.

$- \frac{4 (2 {(0)}^{6} - 64 {(0)}^{4} + 512 {(0)}^{2} - 3 {(0)}^{2} | {(0)}^{4} - 32 {(0)}^{2} + 256 | + 16 | {(0)}^{4} - 32 {(0)}^{2} + 256 |)}{{| {(0)}^{2} - 16 |}^{2} | ((0) + 4) ((0) - 4) |}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$- \frac{4 (2 \cdot 0 - 64 \cdot 0^{4} + 512 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.2

Multiply $2$ by $0$ .

$- \frac{4 (0 - 64 \cdot 0^{4} + 512 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.3

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

$- \frac{4 (0 - 64 \cdot 0 + 512 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.4

Multiply $- 64$ by $0$ .

$- \frac{4 (0 + 0 + 512 \cdot 0^{2} - 3 \cdot 0^{2} | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.5

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

$- \frac{4 (0 + 0 + 512 \cdot 0 - 3 \cdot 0^{2} | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.6

Multiply $512$ by $0$ .

$- \frac{4 (0 + 0 + 0 - 3 \cdot 0^{2} | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.7

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

$- \frac{4 (0 + 0 + 0 - 3 \cdot 0 | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.8

Multiply $- 3$ by $0$ .

$- \frac{4 (0 + 0 + 0 + 0 | 0^{4} - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.9

Simplify each term.

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Step 10.1.9.1

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

$- \frac{4 (0 + 0 + 0 + 0 | 0 - 32 \cdot 0^{2} + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.9.2

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

$- \frac{4 (0 + 0 + 0 + 0 | 0 - 32 \cdot 0 + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.9.3

Multiply $- 32$ by $0$ .

$- \frac{4 (0 + 0 + 0 + 0 | 0 + 0 + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.10

Add $0$ and $0$ .

$- \frac{4 (0 + 0 + 0 + 0 | 0 + 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.11

Add $0$ and $256$ .

$- \frac{4 (0 + 0 + 0 + 0 | 256 | + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.12

The absolute value is the distance between a number and zero. The distance between $0$ and $256$ is $256$ .

$- \frac{4 (0 + 0 + 0 + 0 \cdot 256 + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.13

Multiply $0$ by $256$ .

$- \frac{4 (0 + 0 + 0 + 0 + 16 | 0^{4} - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.14

Simplify each term.

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Step 10.1.14.1

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

$- \frac{4 (0 + 0 + 0 + 0 + 16 | 0 - 32 \cdot 0^{2} + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.14.2

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

$- \frac{4 (0 + 0 + 0 + 0 + 16 | 0 - 32 \cdot 0 + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.14.3

Multiply $- 32$ by $0$ .

$- \frac{4 (0 + 0 + 0 + 0 + 16 | 0 + 0 + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.15

Add $0$ and $0$ .

$- \frac{4 (0 + 0 + 0 + 0 + 16 | 0 + 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.16

Add $0$ and $256$ .

$- \frac{4 (0 + 0 + 0 + 0 + 16 | 256 |)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.17

The absolute value is the distance between a number and zero. The distance between $0$ and $256$ is $256$ .

$- \frac{4 (0 + 0 + 0 + 0 + 16 \cdot 256)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.18

Multiply $16$ by $256$ .

$- \frac{4 (0 + 0 + 0 + 0 + 4096)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.19

Add $0$ and $0$ .

$- \frac{4 (0 + 0 + 0 + 4096)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.20

Add $0$ and $0$ .

$- \frac{4 (0 + 0 + 4096)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.21

Add $0$ and $0$ .

$- \frac{4 (0 + 4096)}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.1.22

Add $0$ and $4096$ .

$- \frac{4 \cdot 4096}{{| 0^{2} - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

$- \frac{4 \cdot 4096}{{| 0 - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.2.2

Subtract $16$ from $0$ .

$- \frac{4 \cdot 4096}{{| - 16 |}^{2} | (0 + 4) (0 - 4) |}$

Step 10.2.3

Add $0$ and $4$ .

$- \frac{4 \cdot 4096}{{| - 16 |}^{2} | 4 (0 - 4) |}$

Step 10.2.4

Subtract $4$ from $0$ .

$- \frac{4 \cdot 4096}{{| - 16 |}^{2} | 4 \cdot - 4 |}$

Step 10.2.5

The absolute value is the distance between a number and zero. The distance between $- 16$ and $0$ is $16$ .

$- \frac{4 \cdot 4096}{16^{2} | 4 \cdot - 4 |}$

Step 10.2.6

Raise $16$ to the power of $2$ .

$- \frac{4 \cdot 4096}{256 | 4 \cdot - 4 |}$

Step 10.2.7

Multiply $4$ by $- 4$ .

$- \frac{4 \cdot 4096}{256 | - 16 |}$

Step 10.2.8

The absolute value is the distance between a number and zero. The distance between $- 16$ and $0$ is $16$ .

$- \frac{4 \cdot 4096}{256 \cdot 16}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $4$ by $4096$ .

$- \frac{16384}{256 \cdot 16}$

Step 10.3.2

Multiply $256$ by $16$ .

$- \frac{16384}{4096}$

Step 10.3.3

Divide $16384$ by $4096$ .

$- 1 \cdot 4$

Step 10.3.4

Multiply $- 1$ by $4$ .

$- 4$

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.

$f (0) = 2 | {(0)}^{2} - 16 |$

Step 12.2

Simplify the result.

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Step 12.2.1

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

$f (0) = 2 | 0 - 16 |$

Step 12.2.2

Subtract $16$ from $0$ .

$f (0) = 2 | - 16 |$

Step 12.2.3

The absolute value is the distance between a number and zero. The distance between $- 16$ and $0$ is $16$ .

$f (0) = 2 \cdot 16$

Step 12.2.4

Multiply $2$ by $16$ .

$f (0) = 32$

Step 12.2.5

The final answer is $32$ .

$y = 32$

Step 13

Evaluate the second derivative at $x = - 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{{| {(- 4)}^{2} - 16 |}^{2} | ((- 4) + 4) ((- 4) - 4) |}$

Step 14

Evaluate the second derivative.

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Step 14.1

Raise $- 4$ to the power of $2$ .

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{{| 16 - 16 |}^{2} | ((- 4) + 4) ((- 4) - 4) |}$

Step 14.2

Subtract $16$ from $16$ .

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{{| 0 |}^{2} | ((- 4) + 4) ((- 4) - 4) |}$

Step 14.3

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

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{0^{2} | ((- 4) + 4) ((- 4) - 4) |}$

Step 14.4

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

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{0 | ((- 4) + 4) ((- 4) - 4) |}$

Step 14.5

Add $- 4$ and $4$ .

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{0 | 0 ((- 4) - 4) |}$

Step 14.6

Subtract $4$ from $- 4$ .

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{0 | 0 \cdot - 8 |}$

Step 14.7

Multiply $0$ by $- 8$ .

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{0 | 0 |}$

Step 14.8

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

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{0 \cdot 0}$

Step 14.9

Multiply $0$ by $0$ .

$- \frac{4 (2 {(- 4)}^{6} - 64 {(- 4)}^{4} + 512 {(- 4)}^{2} - 3 {(- 4)}^{2} | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 | + 16 | {(- 4)}^{4} - 32 {(- 4)}^{2} + 256 |)}{0}$

Step 14.10

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.

$(- \infty, - 4) \cup (- 4, 0) \cup (0, 4) \cup (4, \infty)$

Step 15.2

Substitute any number, such as $- 6$ , from the interval $(- \infty, - 4)$ in the first derivative $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$ to check if the result is negative or positive.

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Step 15.2.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = \frac{4 {(- 6)}^{3} - 64 \cdot - 6}{| {(- 6)}^{2} - 16 |}$

Step 15.2.2

Simplify the result.

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Step 15.2.2.1

Simplify the numerator.

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Step 15.2.2.1.1

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

$f' (- 6) = \frac{4 \cdot - 216 - 64 \cdot - 6}{| {(- 6)}^{2} - 16 |}$

Step 15.2.2.1.2

Multiply $4$ by $- 216$ .

$f' (- 6) = \frac{- 864 - 64 \cdot - 6}{| {(- 6)}^{2} - 16 |}$

Step 15.2.2.1.3

Multiply $- 64$ by $- 6$ .

$f' (- 6) = \frac{- 864 + 384}{| {(- 6)}^{2} - 16 |}$

Step 15.2.2.1.4

Add $- 864$ and $384$ .

$f' (- 6) = \frac{- 480}{| {(- 6)}^{2} - 16 |}$

Step 15.2.2.2

Simplify the denominator.

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Step 15.2.2.2.1

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

$f' (- 6) = \frac{- 480}{| 36 - 16 |}$

Step 15.2.2.2.2

Subtract $16$ from $36$ .

$f' (- 6) = \frac{- 480}{| 20 |}$

Step 15.2.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $20$ is $20$ .

$f' (- 6) = \frac{- 480}{20}$

Step 15.2.2.3

Divide $- 480$ by $20$ .

$f' (- 6) = - 24$

Step 15.2.2.4

The final answer is $- 24$ .

$- 24$

Step 15.3

Substitute any number, such as $- 2$ , from the interval $(- 4, 0)$ in the first derivative $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$ to check if the result is negative or positive.

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Step 15.3.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 15.3.2

Simplify the result.

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Step 15.3.2.1

Simplify the numerator.

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Step 15.3.2.1.1

Raise $- 2$ to the power of $3$ .

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

Step 15.3.2.1.2

Multiply $4$ by $- 8$ .

$f' (- 2) = \frac{- 32 - 64 \cdot - 2}{| {(- 2)}^{2} - 16 |}$

Step 15.3.2.1.3

Multiply $- 64$ by $- 2$ .

$f' (- 2) = \frac{- 32 + 128}{| {(- 2)}^{2} - 16 |}$

Step 15.3.2.1.4

Add $- 32$ and $128$ .

$f' (- 2) = \frac{96}{| {(- 2)}^{2} - 16 |}$

Step 15.3.2.2

Simplify the denominator.

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Step 15.3.2.2.1

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

$f' (- 2) = \frac{96}{| 4 - 16 |}$

Step 15.3.2.2.2

Subtract $16$ from $4$ .

$f' (- 2) = \frac{96}{| - 12 |}$

Step 15.3.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 12$ and $0$ is $12$ .

$f' (- 2) = \frac{96}{12}$

Step 15.3.2.3

Divide $96$ by $12$ .

$f' (- 2) = 8$

Step 15.3.2.4

The final answer is $8$ .

$8$

Step 15.4

Substitute any number, such as $2$ , from the interval $(0, 4)$ in the first derivative $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$ to check if the result is negative or positive.

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Step 15.4.1

Replace the variable $x$ with $2$ in the expression.

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

Step 15.4.2

Simplify the result.

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Step 15.4.2.1

Simplify the numerator.

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Step 15.4.2.1.1

Raise $2$ to the power of $3$ .

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

Step 15.4.2.1.2

Multiply $4$ by $8$ .

$f' (2) = \frac{32 - 64 \cdot 2}{| 2^{2} - 16 |}$

Step 15.4.2.1.3

Multiply $- 64$ by $2$ .

$f' (2) = \frac{32 - 128}{| 2^{2} - 16 |}$

Step 15.4.2.1.4

Subtract $128$ from $32$ .

$f' (2) = \frac{- 96}{| 2^{2} - 16 |}$

Step 15.4.2.2

Simplify the denominator.

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Step 15.4.2.2.1

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

$f' (2) = \frac{- 96}{| 4 - 16 |}$

Step 15.4.2.2.2

Subtract $16$ from $4$ .

$f' (2) = \frac{- 96}{| - 12 |}$

Step 15.4.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 12$ and $0$ is $12$ .

$f' (2) = \frac{- 96}{12}$

Step 15.4.2.3

Divide $- 96$ by $12$ .

$f' (2) = - 8$

Step 15.4.2.4

The final answer is $- 8$ .

$- 8$

Step 15.5

Substitute any number, such as $6$ , from the interval $(4, \infty)$ in the first derivative $\frac{4 x^{3} - 64 x}{| x^{2} - 16 |}$ to check if the result is negative or positive.

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Step 15.5.1

Replace the variable $x$ with $6$ in the expression.

$f' (6) = \frac{4 {(6)}^{3} - 64 \cdot 6}{| {(6)}^{2} - 16 |}$

Step 15.5.2

Simplify the result.

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Step 15.5.2.1

Simplify the numerator.

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Step 15.5.2.1.1

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

$f' (6) = \frac{4 \cdot 216 - 64 \cdot 6}{| 6^{2} - 16 |}$

Step 15.5.2.1.2

Multiply $4$ by $216$ .

$f' (6) = \frac{864 - 64 \cdot 6}{| 6^{2} - 16 |}$

Step 15.5.2.1.3

Multiply $- 64$ by $6$ .

$f' (6) = \frac{864 - 384}{| 6^{2} - 16 |}$

Step 15.5.2.1.4

Subtract $384$ from $864$ .

$f' (6) = \frac{480}{| 6^{2} - 16 |}$

Step 15.5.2.2

Simplify the denominator.

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Step 15.5.2.2.1

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

$f' (6) = \frac{480}{| 36 - 16 |}$

Step 15.5.2.2.2

Subtract $16$ from $36$ .

$f' (6) = \frac{480}{| 20 |}$

Step 15.5.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $20$ is $20$ .

$f' (6) = \frac{480}{20}$

Step 15.5.2.3

Divide $480$ by $20$ .

$f' (6) = 24$

Step 15.5.2.4

The final answer is $24$ .

$24$

Step 15.6

Since the first derivative changed signs from negative to positive around $x = - 4$ , then $x = - 4$ is a local minimum.

$x = - 4$ 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 $x = 4$ , then $x = 4$ is a local minimum.

$x = 4$ is a local minimum

Step 15.9

These are the local extrema for $f (x) = 2 | x^{2} - 16 |$ .

$x = - 4$ is a local minimum

$x = 0$ is a local maximum

$x = 4$ is a local minimum

$x = - 4$ is a local minimum

$x = 0$ is a local maximum

$x = 4$ is a local minimum

Step 16