Calculus Examples

$f (x) = \frac{x^{3}}{x^{2} - 16}$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

Move $3$ to the left of $x^{2} - 16$ .

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

Simplify the expression.

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Step 1.1.1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.1.3

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

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

Step 1.1.1.4

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

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

Step 1.1.1.5

Add $1$ and $3$ .

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

Step 1.1.1.6

Simplify.

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Step 1.1.1.6.1

Apply the distributive property.

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

Step 1.1.1.6.2

Apply the distributive property.

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

Step 1.1.1.6.3

Simplify the numerator.

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Step 1.1.1.6.3.1

Simplify each term.

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Step 1.1.1.6.3.1.1

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

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Step 1.1.1.6.3.1.1.1

Move $x^{2}$ .

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

Step 1.1.1.6.3.1.1.2

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

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

Step 1.1.1.6.3.1.1.3

Add $2$ and $2$ .

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

Step 1.1.1.6.3.1.2

Multiply $3$ by $- 16$ .

$f' (x) = \frac{3 x^{4} - 48 x^{2} - 2 x^{4}}{{(x^{2} - 16)}^{2}}$

Step 1.1.1.6.3.2

Subtract $2 x^{4}$ from $3 x^{4}$ .

$f' (x) = \frac{x^{4} - 48 x^{2}}{{(x^{2} - 16)}^{2}}$

Step 1.1.1.6.4

Factor $x^{2}$ out of $x^{4} - 48 x^{2}$ .

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Step 1.1.1.6.4.1

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

$f' (x) = \frac{x^{2} x^{2} - 48 x^{2}}{{(x^{2} - 16)}^{2}}$

Step 1.1.1.6.4.2

Factor $x^{2}$ out of $- 48 x^{2}$ .

$f' (x) = \frac{x^{2} x^{2} + x^{2} \cdot - 48}{{(x^{2} - 16)}^{2}}$

Step 1.1.1.6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 48$ .

$f' (x) = \frac{x^{2} (x^{2} - 48)}{{(x^{2} - 16)}^{2}}$

Step 1.1.1.6.5

Simplify the denominator.

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Step 1.1.1.6.5.1

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

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

Step 1.1.1.6.5.2

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 $b = 4$ .

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

Step 1.1.1.6.5.3

Apply the product rule to $(x + 4) (x - 4)$ .

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

Step 1.1.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 48$ .

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

Step 1.1.2.3

Differentiate.

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Step 1.1.2.3.1

By the Sum Rule, the derivative of $x^{2} - 48$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 48]$ .

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

Step 1.1.2.3.2

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

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

Step 1.1.2.3.3

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (x^{2} (2 x + 0) + (x^{2} - 48) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 48) \frac{d}{d x} ({(x + 4)}^{2} {(x - 4)}^{2})}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.3.4

Add $2 x$ and $0$ .

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

Step 1.1.2.4

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

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Step 1.1.2.4.1

Move $x$ .

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

Step 1.1.2.4.2

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

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Step 1.1.2.4.2.1

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

Step 1.1.2.4.2.2

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (x^{1 + 2} \cdot 2 + (x^{2} - 48) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 48) \frac{d}{d x} ({(x + 4)}^{2} {(x - 4)}^{2})}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.4.3

Add $1$ and $2$ .

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

Step 1.1.2.5

Differentiate using the Power Rule.

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Step 1.1.2.5.1

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

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

Step 1.1.2.5.2

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + (x^{2} - 48) (2 x)) - x^{2} (x^{2} - 48) \frac{d}{d x} ({(x + 4)}^{2} {(x - 4)}^{2})}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.5.3

Move $2$ to the left of $x^{2} - 48$ .

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

Step 1.1.2.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(x + 4)}^{2}$ and $g (x) = {(x - 4)}^{2}$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) ({(x + 4)}^{2} \frac{d}{d x} ({(x - 4)}^{2}) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.7

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

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Step 1.1.2.7.1

To apply the Chain Rule, set $u_{1}$ as $x - 4$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) ({(x + 4)}^{2} (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (x - 4)) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.7.2

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) ({(x + 4)}^{2} (2 u_{1} \frac{d}{d x} (x - 4)) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.7.3

Replace all occurrences of $u_{1}$ with $x - 4$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) ({(x + 4)}^{2} (2 (x - 4) \frac{d}{d x} (x - 4)) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.8

Differentiate.

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Step 1.1.2.8.1

Move $2$ to the left of ${(x + 4)}^{2}$ .

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

Step 1.1.2.8.2

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) (\frac{d}{d x} (x) + \frac{d}{d x} (- 4)) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.8.3

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) (1 + \frac{d}{d x} (- 4)) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.8.4

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) (1 + 0) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.8.5

Simplify the expression.

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Step 1.1.2.8.5.1

Add $1$ and $0$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) \cdot 1 + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.8.5.2

Multiply $2$ by $1$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + {(x - 4)}^{2} \frac{d}{d x} ({(x + 4)}^{2}))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.9

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

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Step 1.1.2.9.1

To apply the Chain Rule, set $u_{2}$ as $x + 4$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + {(x - 4)}^{2} (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (x + 4)))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.9.2

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + {(x - 4)}^{2} (2 u_{2} \frac{d}{d x} (x + 4)))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.9.3

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + {(x - 4)}^{2} (2 (x + 4) \frac{d}{d x} (x + 4)))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.10

Differentiate.

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Step 1.1.2.10.1

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

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

Step 1.1.2.10.2

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4) (\frac{d}{d x} (x) + \frac{d}{d x} (4)))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.10.3

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4) (1 + \frac{d}{d x} (4)))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.10.4

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

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4) (1 + 0))}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.10.5

Simplify the expression.

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Step 1.1.2.10.5.1

Add $1$ and $0$ .

$f'' (x) = \frac{{(x + 4)}^{2} {(x - 4)}^{2} (2 x^{3} + 2 (x^{2} - 48) x) - x^{2} (x^{2} - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4) \cdot 1)}{{({(x + 4)}^{2} {(x - 4)}^{2})}^{2}}$

Step 1.1.2.10.5.2

Multiply $2$ by $1$ .

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

Step 1.1.2.11

Simplify.

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Step 1.1.2.11.1

Apply the product rule to ${(x + 4)}^{2} {(x - 4)}^{2}$ .

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

Step 1.1.2.11.2

Apply the distributive property.

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

Step 1.1.2.11.3

Apply the distributive property.

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

Step 1.1.2.11.4

Apply the distributive property.

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

Step 1.1.2.11.5

Simplify the numerator.

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Step 1.1.2.11.5.1

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

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

Step 1.1.2.11.5.2

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

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Step 1.1.2.11.5.2.1

Apply the distributive property.

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

Step 1.1.2.11.5.2.2

Apply the distributive property.

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

Step 1.1.2.11.5.2.3

Apply the distributive property.

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

Step 1.1.2.11.5.3

Simplify and combine like terms.

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Step 1.1.2.11.5.3.1

Simplify each term.

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Step 1.1.2.11.5.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.2.11.5.3.1.2

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

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

Step 1.1.2.11.5.3.1.3

Multiply $4$ by $4$ .

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

Step 1.1.2.11.5.3.2

Add $4 x$ and $4 x$ .

$f'' (x) = \frac{(x^{2} + 8 x + 16) {(x - 4)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.4

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

$f'' (x) = \frac{(x^{2} + 8 x + 16) ((x - 4) (x - 4)) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.5

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

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Step 1.1.2.11.5.5.1

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 8 x + 16) (x (x - 4) - 4 (x - 4)) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.5.2

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 8 x + 16) (x \cdot x + x \cdot - 4 - 4 (x - 4)) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.5.3

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 8 x + 16) (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.6

Simplify and combine like terms.

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Step 1.1.2.11.5.6.1

Simplify each term.

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Step 1.1.2.11.5.6.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{(x^{2} + 8 x + 16) (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.6.1.2

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

$f'' (x) = \frac{(x^{2} + 8 x + 16) (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.6.1.3

Multiply $- 4$ by $- 4$ .

$f'' (x) = \frac{(x^{2} + 8 x + 16) (x^{2} - 4 x - 4 x + 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.6.2

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

$f'' (x) = \frac{(x^{2} + 8 x + 16) (x^{2} - 8 x + 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.7

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

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} (- 8 x) + x^{2} \cdot 16 + 8 x \cdot x^{2} + 8 x (- 8 x) + 8 x \cdot 16 + 16 x^{2} + 16 (- 8 x) + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.8

Combine the opposite terms in $x^{2} x^{2} + x^{2} (- 8 x) + x^{2} \cdot 16 + 8 x \cdot x^{2} + 8 x (- 8 x) + 8 x \cdot 16 + 16 x^{2} + 16 (- 8 x) + 16 \cdot 16$ .

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Step 1.1.2.11.5.8.1

Reorder the factors in the terms $x^{2} (- 8 x)$ and $8 x \cdot x^{2}$ .

$f'' (x) = \frac{(x^{2} x^{2} - 8 x^{2} x + x^{2} \cdot 16 + 8 x^{2} x + 8 x (- 8 x) + 8 x \cdot 16 + 16 x^{2} + 16 (- 8 x) + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.8.2

Add $- 8 x^{2} x$ and $8 x^{2} x$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 16 + 0 + 8 x (- 8 x) + 8 x \cdot 16 + 16 x^{2} + 16 (- 8 x) + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.8.3

Add $x^{2} x^{2} + x^{2} \cdot 16$ and $0$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 16 + 8 x (- 8 x) + 8 x \cdot 16 + 16 x^{2} + 16 (- 8 x) + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.8.4

Reorder the factors in the terms $8 x \cdot 16$ and $16 (- 8 x)$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 16 + 8 x (- 8 x) + 8 \cdot (16 x) + 16 x^{2} - 8 \cdot (16 x) + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.8.5

Subtract $8 \cdot 16 x$ from $8 \cdot 16 x$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 16 + 8 x (- 8 x) + 16 x^{2} + 0 + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.8.6

Add $x^{2} x^{2} + x^{2} \cdot 16 + 8 x (- 8 x) + 16 x^{2}$ and $0$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 16 + 8 x (- 8 x) + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9

Simplify each term.

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Step 1.1.2.11.5.9.1

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

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Step 1.1.2.11.5.9.1.1

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

$f'' (x) = \frac{(x^{2 + 2} + x^{2} \cdot 16 + 8 x (- 8 x) + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{(x^{4} + x^{2} \cdot 16 + 8 x (- 8 x) + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9.2

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

$f'' (x) = \frac{(x^{4} + 16 \cdot x^{2} + 8 x (- 8 x) + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{(x^{4} + 16 x^{2} + 8 \cdot (- 8 x \cdot x) + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9.4

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

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Step 1.1.2.11.5.9.4.1

Move $x$ .

$f'' (x) = \frac{(x^{4} + 16 x^{2} + 8 \cdot (- 8 (x \cdot x)) + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9.4.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{(x^{4} + 16 x^{2} + 8 \cdot (- 8 x^{2}) + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9.5

Multiply $8$ by $- 8$ .

$f'' (x) = \frac{(x^{4} + 16 x^{2} - 64 x^{2} + 16 x^{2} + 16 \cdot 16) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.9.6

Multiply $16$ by $16$ .

$f'' (x) = \frac{(x^{4} + 16 x^{2} - 64 x^{2} + 16 x^{2} + 256) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.10

Subtract $64 x^{2}$ from $16 x^{2}$ .

$f'' (x) = \frac{(x^{4} - 48 x^{2} + 16 x^{2} + 256) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.11

Add $- 48 x^{2}$ and $16 x^{2}$ .

$f'' (x) = \frac{(x^{4} - 32 x^{2} + 256) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.12

Simplify each term.

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Step 1.1.2.11.5.12.1

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

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Step 1.1.2.11.5.12.1.1

Move $x$ .

$f'' (x) = \frac{(x^{4} - 32 x^{2} + 256) (2 x^{3} + 2 (x \cdot x^{2}) + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.12.1.2

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

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Step 1.1.2.11.5.12.1.2.1

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

Step 1.1.2.11.5.12.1.2.2

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

$f'' (x) = \frac{(x^{4} - 32 x^{2} + 256) (2 x^{3} + 2 x^{1 + 2} + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.12.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{(x^{4} - 32 x^{2} + 256) (2 x^{3} + 2 x^{3} + 2 \cdot (- 48 x)) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.12.2

Multiply $2$ by $- 48$ .

$f'' (x) = \frac{(x^{4} - 32 x^{2} + 256) (2 x^{3} + 2 x^{3} - 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.13

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

$f'' (x) = \frac{(x^{4} - 32 x^{2} + 256) (4 x^{3} - 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.14

Expand $(x^{4} - 32 x^{2} + 256) (4 x^{3} - 96 x)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{x^{4} (4 x^{3}) + x^{4} (- 96 x) - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15

Simplify each term.

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Step 1.1.2.11.5.15.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{4} x^{3} + x^{4} (- 96 x) - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 1.1.2.11.5.15.2.1

Move $x^{3}$ .

$f'' (x) = \frac{4 (x^{3} x^{4}) + x^{4} (- 96 x) - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.2.2

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

$f'' (x) = \frac{4 x^{3 + 4} + x^{4} (- 96 x) - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{4 x^{7} + x^{4} (- 96 x) - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 96 x^{4} x - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.4

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

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Step 1.1.2.11.5.15.4.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 96 (x \cdot x^{4}) - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.4.2

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

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Step 1.1.2.11.5.15.4.2.1

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

Step 1.1.2.11.5.15.4.2.2

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

$f'' (x) = \frac{4 x^{7} - 96 x^{1 + 4} - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.4.3

Add $1$ and $4$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 32 x^{2} (4 x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 32 \cdot (4 x^{2} x^{3}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.6

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

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Step 1.1.2.11.5.15.6.1

Move $x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 32 \cdot (4 (x^{3} x^{2})) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.6.2

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

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 32 \cdot (4 x^{3 + 2}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.6.3

Add $3$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 32 \cdot (4 x^{5}) - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.7

Multiply $- 32$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} - 32 x^{2} (- 96 x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.8

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} - 32 \cdot (- 96 x^{2} x) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.9

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

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Step 1.1.2.11.5.15.9.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} - 32 \cdot (- 96 (x \cdot x^{2})) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.9.2

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

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Step 1.1.2.11.5.15.9.2.1

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

Step 1.1.2.11.5.15.9.2.2

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

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} - 32 \cdot (- 96 x^{1 + 2}) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.9.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} - 32 \cdot (- 96 x^{3}) + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.10

Multiply $- 32$ by $- 96$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} + 3072 x^{3} + 256 (4 x^{3}) + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.11

Multiply $4$ by $256$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} + 3072 x^{3} + 1024 x^{3} + 256 (- 96 x) + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.15.12

Multiply $- 96$ by $256$ .

$f'' (x) = \frac{4 x^{7} - 96 x^{5} - 128 x^{5} + 3072 x^{3} + 1024 x^{3} - 24576 x + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.16

Subtract $128 x^{5}$ from $- 96 x^{5}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 3072 x^{3} + 1024 x^{3} - 24576 x + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.17

Add $3072 x^{3}$ and $1024 x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{2} x^{2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.18

Simplify each term.

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Step 1.1.2.11.5.18.1

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

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Step 1.1.2.11.5.18.1.1

Move $x^{2}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- (x^{2} x^{2}) - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.18.1.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{2 + 2} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.18.1.3

Add $2$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} - x^{2} \cdot - 48) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.18.2

Multiply $- 48$ by $- 1$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 {(x + 4)}^{2} (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19

Simplify each term.

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Step 1.1.2.11.5.19.1

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 ((x + 4) (x + 4)) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.2

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

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Step 1.1.2.11.5.19.2.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x (x + 4) + 4 (x + 4)) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.2.2

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x \cdot x + x \cdot 4 + 4 (x + 4)) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.2.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.3

Simplify and combine like terms.

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Step 1.1.2.11.5.19.3.1

Simplify each term.

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Step 1.1.2.11.5.19.3.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.3.1.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.3.1.3

Multiply $4$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x^{2} + 4 x + 4 x + 16) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.3.2

Add $4 x$ and $4 x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x^{2} + 8 x + 16) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.4

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) ((2 x^{2} + 2 (8 x) + 2 \cdot 16) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.5

Simplify.

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Step 1.1.2.11.5.19.5.1

Multiply $8$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) ((2 x^{2} + 16 x + 2 \cdot 16) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.5.2

Multiply $2$ by $16$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) ((2 x^{2} + 16 x + 32) (x - 4) + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.6

Expand $(2 x^{2} + 16 x + 32) (x - 4)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{2} x + 2 x^{2} \cdot - 4 + 16 x \cdot x + 16 x \cdot - 4 + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7

Simplify each term.

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Step 1.1.2.11.5.19.7.1

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

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Step 1.1.2.11.5.19.7.1.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 (x \cdot x^{2}) + 2 x^{2} \cdot - 4 + 16 x \cdot x + 16 x \cdot - 4 + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7.1.2

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

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Step 1.1.2.11.5.19.7.1.2.1

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

Step 1.1.2.11.5.19.7.1.2.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{1 + 2} + 2 x^{2} \cdot - 4 + 16 x \cdot x + 16 x \cdot - 4 + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 2 x^{2} \cdot - 4 + 16 x \cdot x + 16 x \cdot - 4 + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7.2

Multiply $- 4$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 8 x^{2} + 16 x \cdot x + 16 x \cdot - 4 + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7.3

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

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Step 1.1.2.11.5.19.7.3.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 8 x^{2} + 16 (x \cdot x) + 16 x \cdot - 4 + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 8 x^{2} + 16 x^{2} + 16 x \cdot - 4 + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7.4

Multiply $- 4$ by $16$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 8 x^{2} + 16 x^{2} - 64 x + 32 x + 32 \cdot - 4 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.7.5

Multiply $32$ by $- 4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 8 x^{2} + 16 x^{2} - 64 x + 32 x - 128 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.8

Add $- 8 x^{2}$ and $16 x^{2}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 64 x + 32 x - 128 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.9

Add $- 64 x$ and $32 x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 {(x - 4)}^{2} (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.10

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 ((x - 4) (x - 4)) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.11

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

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Step 1.1.2.11.5.19.11.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x (x - 4) - 4 (x - 4)) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.11.2

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x \cdot x + x \cdot - 4 - 4 (x - 4)) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.11.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.12

Simplify and combine like terms.

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Step 1.1.2.11.5.19.12.1

Simplify each term.

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Step 1.1.2.11.5.19.12.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.12.1.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.12.1.3

Multiply $- 4$ by $- 4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x^{2} - 4 x - 4 x + 16) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.12.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x^{2} - 8 x + 16) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.13

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + (2 x^{2} + 2 (- 8 x) + 2 \cdot 16) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.14

Simplify.

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Step 1.1.2.11.5.19.14.1

Multiply $- 8$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + (2 x^{2} - 16 x + 2 \cdot 16) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.14.2

Multiply $2$ by $16$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + (2 x^{2} - 16 x + 32) (x + 4))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.15

Expand $(2 x^{2} - 16 x + 32) (x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{2} x + 2 x^{2} \cdot 4 - 16 x \cdot x - 16 x \cdot 4 + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16

Simplify each term.

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Step 1.1.2.11.5.19.16.1

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

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Step 1.1.2.11.5.19.16.1.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 (x \cdot x^{2}) + 2 x^{2} \cdot 4 - 16 x \cdot x - 16 x \cdot 4 + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16.1.2

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

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Step 1.1.2.11.5.19.16.1.2.1

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

Step 1.1.2.11.5.19.16.1.2.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{1 + 2} + 2 x^{2} \cdot 4 - 16 x \cdot x - 16 x \cdot 4 + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} + 2 x^{2} \cdot 4 - 16 x \cdot x - 16 x \cdot 4 + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16.2

Multiply $4$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} + 8 x^{2} - 16 x \cdot x - 16 x \cdot 4 + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16.3

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

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Step 1.1.2.11.5.19.16.3.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} + 8 x^{2} - 16 (x \cdot x) - 16 x \cdot 4 + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} + 8 x^{2} - 16 x^{2} - 16 x \cdot 4 + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16.4

Multiply $4$ by $- 16$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} + 8 x^{2} - 16 x^{2} - 64 x + 32 x + 32 \cdot 4)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.16.5

Multiply $32$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} + 8 x^{2} - 16 x^{2} - 64 x + 32 x + 128)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.17

Subtract $16 x^{2}$ from $8 x^{2}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} - 8 x^{2} - 64 x + 32 x + 128)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.19.18

Add $- 64 x$ and $32 x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} - 8 x^{2} - 32 x + 128)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.20

Combine the opposite terms in $2 x^{3} + 8 x^{2} - 32 x - 128 + 2 x^{3} - 8 x^{2} - 32 x + 128$ .

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Step 1.1.2.11.5.20.1

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 32 x - 128 + 2 x^{3} + 0 - 32 x + 128)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.20.2

Add $2 x^{3} - 32 x - 128 + 2 x^{3}$ and $0$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 32 x - 128 + 2 x^{3} - 32 x + 128)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.20.3

Add $- 128$ and $128$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 32 x + 2 x^{3} - 32 x + 0)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.20.4

Add $2 x^{3} - 32 x + 2 x^{3} - 32 x$ and $0$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (2 x^{3} - 32 x + 2 x^{3} - 32 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.21

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (4 x^{3} - 32 x - 32 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.22

Subtract $32 x$ from $- 32 x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x + (- x^{4} + 48 x^{2}) (4 x^{3} - 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.23

Expand $(- x^{4} + 48 x^{2}) (4 x^{3} - 64 x)$ using the FOIL Method.

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Step 1.1.2.11.5.23.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - x^{4} (4 x^{3} - 64 x) + 48 x^{2} (4 x^{3} - 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.23.2

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - x^{4} (4 x^{3}) - x^{4} (- 64 x) + 48 x^{2} (4 x^{3} - 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.23.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - x^{4} (4 x^{3}) - x^{4} (- 64 x) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24

Simplify and combine like terms.

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Step 1.1.2.11.5.24.1

Simplify each term.

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Step 1.1.2.11.5.24.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 1 \cdot (4 x^{4} x^{3}) - x^{4} (- 64 x) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 1.1.2.11.5.24.1.2.1

Move $x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 1 \cdot (4 (x^{3} x^{4})) - x^{4} (- 64 x) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.2.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 1 \cdot (4 x^{3 + 4}) - x^{4} (- 64 x) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 1 \cdot (4 x^{7}) - x^{4} (- 64 x) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.3

Multiply $- 1$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} - x^{4} (- 64 x) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} - 1 \cdot (- 64 x^{4} x) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.5

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

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Step 1.1.2.11.5.24.1.5.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} - 1 \cdot (- 64 (x \cdot x^{4})) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.5.2

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

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Step 1.1.2.11.5.24.1.5.2.1

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} - 1 \cdot (- 64 (x \cdot x^{4})) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.5.2.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} - 1 \cdot (- 64 x^{1 + 4}) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.5.3

Add $1$ and $4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} - 1 \cdot (- 64 x^{5}) + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.6

Multiply $- 1$ by $- 64$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 48 x^{2} (4 x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 48 \cdot (4 x^{2} x^{3}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.8

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

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Step 1.1.2.11.5.24.1.8.1

Move $x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 48 \cdot (4 (x^{3} x^{2})) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.8.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 48 \cdot (4 x^{3 + 2}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.8.3

Add $3$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 48 \cdot (4 x^{5}) + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.9

Multiply $48$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 192 x^{5} + 48 x^{2} (- 64 x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.10

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 192 x^{5} + 48 \cdot (- 64 x^{2} x)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.11

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

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Step 1.1.2.11.5.24.1.11.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 192 x^{5} + 48 \cdot (- 64 (x \cdot x^{2}))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.11.2

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

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Step 1.1.2.11.5.24.1.11.2.1

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 192 x^{5} + 48 \cdot (- 64 (x \cdot x^{2}))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.11.2.2

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

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 192 x^{5} + 48 \cdot (- 64 x^{1 + 2})}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.11.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 192 x^{5} + 48 \cdot (- 64 x^{3})}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.1.12

Multiply $48$ by $- 64$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 64 x^{5} + 192 x^{5} - 3072 x^{3}}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.24.2

Add $64 x^{5}$ and $192 x^{5}$ .

$f'' (x) = \frac{4 x^{7} - 224 x^{5} + 4096 x^{3} - 24576 x - 4 x^{7} + 256 x^{5} - 3072 x^{3}}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.25

Subtract $4 x^{7}$ from $4 x^{7}$ .

$f'' (x) = \frac{- 224 x^{5} + 4096 x^{3} - 24576 x + 0 + 256 x^{5} - 3072 x^{3}}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.26

Add $- 224 x^{5}$ and $0$ .

$f'' (x) = \frac{- 224 x^{5} + 4096 x^{3} - 24576 x + 256 x^{5} - 3072 x^{3}}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.27

Add $- 224 x^{5}$ and $256 x^{5}$ .

$f'' (x) = \frac{32 x^{5} + 4096 x^{3} - 24576 x - 3072 x^{3}}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.28

Subtract $3072 x^{3}$ from $4096 x^{3}$ .

$f'' (x) = \frac{32 x^{5} + 1024 x^{3} - 24576 x}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29

Rewrite $32 x^{5} + 1024 x^{3} - 24576 x$ in a factored form.

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Step 1.1.2.11.5.29.1

Factor $32 x$ out of $32 x^{5} + 1024 x^{3} - 24576 x$ .

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Step 1.1.2.11.5.29.1.1

Factor $32 x$ out of $32 x^{5}$ .

$f'' (x) = \frac{32 x (x^{4}) + 1024 x^{3} - 24576 x}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.1.2

Factor $32 x$ out of $1024 x^{3}$ .

$f'' (x) = \frac{32 x (x^{4}) + 32 x (32 x^{2}) - 24576 x}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.1.3

Factor $32 x$ out of $- 24576 x$ .

$f'' (x) = \frac{32 x (x^{4}) + 32 x (32 x^{2}) + 32 x (- 768)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.1.4

Factor $32 x$ out of $32 x (x^{4}) + 32 x (32 x^{2})$ .

$f'' (x) = \frac{32 x (x^{4} + 32 x^{2}) + 32 x (- 768)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.1.5

Factor $32 x$ out of $32 x (x^{4} + 32 x^{2}) + 32 x (- 768)$ .

$f'' (x) = \frac{32 x (x^{4} + 32 x^{2} - 768)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.2

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

$f'' (x) = \frac{32 x ({(x^{2})}^{2} + 32 x^{2} - 768)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.3

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

$f'' (x) = \frac{32 x ({u_{3}}^{2} + 32 u_{3} - 768)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.4

Factor ${u_{3}}^{2} + 32 u_{3} - 768$ using the AC method.

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Step 1.1.2.11.5.29.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 768$ and whose sum is $32$ .

$- 16, 48$

Step 1.1.2.11.5.29.4.2

Write the factored form using these integers.

$f'' (x) = \frac{32 x ((u_{3} - 16) (u_{3} + 48))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.5

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

$f'' (x) = \frac{32 x ((x^{2} - 16) (x^{2} + 48))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.6

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

$f'' (x) = \frac{32 x ((x^{2} - 4^{2}) (x^{2} + 48))}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.5.29.7

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 $b = 4$ .

$f'' (x) = \frac{32 x (x + 4) (x - 4) (x^{2} + 48)}{{({(x + 4)}^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.6

Combine terms.

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Step 1.1.2.11.6.1

Multiply the exponents in ${({(x + 4)}^{2})}^{2}$ .

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Step 1.1.2.11.6.1.1

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

$f'' (x) = \frac{32 x (x + 4) (x - 4) (x^{2} + 48)}{{(x + 4)}^{2 \cdot 2} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.6.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{32 x (x + 4) (x - 4) (x^{2} + 48)}{{(x + 4)}^{4} {({(x - 4)}^{2})}^{2}}$

Step 1.1.2.11.6.2

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

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Step 1.1.2.11.6.2.1

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

$f'' (x) = \frac{32 x (x + 4) (x - 4) (x^{2} + 48)}{{(x + 4)}^{4} {(x - 4)}^{2 \cdot 2}}$

Step 1.1.2.11.6.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{32 x (x + 4) (x - 4) (x^{2} + 48)}{{(x + 4)}^{4} {(x - 4)}^{4}}$

Step 1.1.2.11.6.3

Cancel the common factor of $x + 4$ and ${(x + 4)}^{4}$ .

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Step 1.1.2.11.6.3.1

Factor $x + 4$ out of $32 x (x + 4) (x - 4) (x^{2} + 48)$ .

$f'' (x) = \frac{(x + 4) ((32 x (x - 4)) (x^{2} + 48))}{{(x + 4)}^{4} {(x - 4)}^{4}}$

Step 1.1.2.11.6.3.2

Cancel the common factors.

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Step 1.1.2.11.6.3.2.1

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

$f'' (x) = \frac{(x + 4) ((32 x (x - 4)) (x^{2} + 48))}{(x + 4) ({(x + 4)}^{3} {(x - 4)}^{4})}$

Step 1.1.2.11.6.3.2.2

Cancel the common factor.

$f'' (x) = \frac{(x + 4) ((32 x (x - 4)) (x^{2} + 48))}{(x + 4) ({(x + 4)}^{3} {(x - 4)}^{4})}$

Step 1.1.2.11.6.3.2.3

Rewrite the expression.

$f'' (x) = \frac{(32 x (x - 4)) (x^{2} + 48)}{{(x + 4)}^{3} {(x - 4)}^{4}}$

Step 1.1.2.11.6.4

Cancel the common factor of $x - 4$ and ${(x - 4)}^{4}$ .

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Step 1.1.2.11.6.4.1

Factor $x - 4$ out of $32 x (x - 4) (x^{2} + 48)$ .

$f'' (x) = \frac{(x - 4) (32 x (x^{2} + 48))}{{(x + 4)}^{3} {(x - 4)}^{4}}$

Step 1.1.2.11.6.4.2

Cancel the common factors.

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Step 1.1.2.11.6.4.2.1

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

$f'' (x) = \frac{(x - 4) (32 x (x^{2} + 48))}{(x - 4) ({(x + 4)}^{3} {(x - 4)}^{3})}$

Step 1.1.2.11.6.4.2.2

Cancel the common factor.

$f'' (x) = \frac{(x - 4) (32 x (x^{2} + 48))}{(x - 4) ({(x + 4)}^{3} {(x - 4)}^{3})}$

Step 1.1.2.11.6.4.2.3

Rewrite the expression.

$f'' (x) = \frac{32 x (x^{2} + 48)}{{(x + 4)}^{3} {(x - 4)}^{3}}$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{32 x (x^{2} + 48)}{{(x + 4)}^{3} {(x - 4)}^{3}}$ .

$\frac{32 x (x^{2} + 48)}{{(x + 4)}^{3} {(x - 4)}^{3}}$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $\frac{32 x (x^{2} + 48)}{{(x + 4)}^{3} {(x - 4)}^{3}} = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$\frac{32 x (x^{2} + 48)}{{(x + 4)}^{3} {(x - 4)}^{3}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$32 x (x^{2} + 48) = 0$

Step 1.2.3

Solve the equation for $x$ .

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Step 1.2.3.1

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

$x = 0$

$x^{2} + 48 = 0$

Step 1.2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3.3

Set $x^{2} + 48$ equal to $0$ and solve for $x$ .

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Step 1.2.3.3.1

Set $x^{2} + 48$ equal to $0$ .

$x^{2} + 48 = 0$

Step 1.2.3.3.2

Solve $x^{2} + 48 = 0$ for $x$ .

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Step 1.2.3.3.2.1

Subtract $48$ from both sides of the equation.

$x^{2} = - 48$

Step 1.2.3.3.2.2

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

$x = \pm \sqrt{- 48}$

Step 1.2.3.3.2.3

Simplify $\pm \sqrt{- 48}$ .

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Step 1.2.3.3.2.3.1

Rewrite $- 48$ as $- 1 (48)$ .

$x = \pm \sqrt{- 1 (48)}$

Step 1.2.3.3.2.3.2

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{48}$

Step 1.2.3.3.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{48}$

Step 1.2.3.3.2.3.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 1.2.3.3.2.3.4.1

Factor $16$ out of $48$ .

$x = \pm i \cdot \sqrt{16 (3)}$

Step 1.2.3.3.2.3.4.2

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

$x = \pm i \cdot \sqrt{4^{2} \cdot 3}$

Step 1.2.3.3.2.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (4 \sqrt{3})$

Step 1.2.3.3.2.3.6

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

$x = \pm 4 i \sqrt{3}$

Step 1.2.3.3.2.4

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

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Step 1.2.3.3.2.4.1

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

$x = 4 i \sqrt{3}$

Step 1.2.3.3.2.4.2

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

$x = - 4 i \sqrt{3}$

Step 1.2.3.3.2.4.3

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

$x = 4 i \sqrt{3}, - 4 i \sqrt{3}$

Step 1.2.3.4

The final solution is all the values that make $32 x (x^{2} + 48) = 0$ true.

$x = 0, 4 i \sqrt{3}, - 4 i \sqrt{3}$

Step 2

Find the domain of $f (x) = \frac{x^{3}}{x^{2} - 16}$ .

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Step 2.1

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

$x^{2} - 16 = 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

Add $16$ to both sides of the equation.

$x^{2} = 16$

Step 2.2.2

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

$x = \pm \sqrt{16}$

Step 2.2.3

Simplify $\pm \sqrt{16}$ .

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

$x = \pm 4$

Step 2.2.4

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

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Step 2.2.4.1

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

$x = 4$

Step 2.2.4.2

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

$x = - 4$

Step 2.2.4.3

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

$x = 4, - 4$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 4, - 4}$

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 4, - 4}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

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

Step 4

Substitute any number from the interval $(- \infty, - 4)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

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

$f'' (- 6) = \frac{32 (- 6) ({(- 6)}^{2} + 48)}{{((- 6) + 4)}^{3} {((- 6) - 4)}^{3}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $32$ by $- 6$ .

$f'' (- 6) = \frac{- 192 ({(- 6)}^{2} + 48)}{{(- 6 + 4)}^{3} {(- 6 - 4)}^{3}}$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Add $- 6$ and $4$ .

$f'' (- 6) = \frac{- 192 ({(- 6)}^{2} + 48)}{{(- 2)}^{3} {(- 6 - 4)}^{3}}$

Step 4.2.2.2

Subtract $4$ from $- 6$ .

$f'' (- 6) = \frac{- 192 ({(- 6)}^{2} + 48)}{{(- 2)}^{3} {(- 10)}^{3}}$

Step 4.2.2.3

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

$f'' (- 6) = \frac{- 192 ({(- 6)}^{2} + 48)}{- 8 {(- 10)}^{3}}$

Step 4.2.2.4

Raise $- 10$ to the power of $3$ .

$f'' (- 6) = \frac{- 192 ({(- 6)}^{2} + 48)}{- 8 \cdot - 1000}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

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

$f'' (- 6) = \frac{- 192 (36 + 48)}{- 8 \cdot - 1000}$

Step 4.2.3.2

Add $36$ and $48$ .

$f'' (- 6) = \frac{- 192 \cdot 84}{- 8 \cdot - 1000}$

Step 4.2.4

Reduce the expression by cancelling the common factors.

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Step 4.2.4.1

Multiply $- 8$ by $- 1000$ .

$f'' (- 6) = \frac{- 192 \cdot 84}{8000}$

Step 4.2.4.2

Multiply $- 192$ by $84$ .

$f'' (- 6) = \frac{- 16128}{8000}$

Step 4.2.4.3

Cancel the common factor of $- 16128$ and $8000$ .

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Step 4.2.4.3.1

Factor $64$ out of $- 16128$ .

$f'' (- 6) = \frac{64 (- 252)}{8000}$

Step 4.2.4.3.2

Cancel the common factors.

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Step 4.2.4.3.2.1

Factor $64$ out of $8000$ .

$f'' (- 6) = \frac{64 \cdot - 252}{64 \cdot 125}$

Step 4.2.4.3.2.2

Cancel the common factor.

$f'' (- 6) = \frac{64 \cdot - 252}{64 \cdot 125}$

Step 4.2.4.3.2.3

Rewrite the expression.

$f'' (- 6) = \frac{- 252}{125}$

Step 4.2.4.4

Move the negative in front of the fraction.

$f'' (- 6) = - \frac{252}{125}$

Step 4.2.5

The final answer is $- \frac{252}{125}$ .

$- \frac{252}{125}$

Step 4.3

The graph is concave down on the interval $(- \infty, - 4)$ because $f'' (- 6)$ is negative.

Concave down on $(- \infty, - 4)$ since $f'' (x)$ is negative

Step 5

Substitute any number from the interval $(- 4, 0)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (- 2) = \frac{32 (- 2) ({(- 2)}^{2} + 48)}{{((- 2) + 4)}^{3} {((- 2) - 4)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $32$ by $- 2$ .

$f'' (- 2) = \frac{- 64 ({(- 2)}^{2} + 48)}{{(- 2 + 4)}^{3} {(- 2 - 4)}^{3}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Add $- 2$ and $4$ .

$f'' (- 2) = \frac{- 64 ({(- 2)}^{2} + 48)}{2^{3} {(- 2 - 4)}^{3}}$

Step 5.2.2.2

Subtract $4$ from $- 2$ .

$f'' (- 2) = \frac{- 64 ({(- 2)}^{2} + 48)}{2^{3} {(- 6)}^{3}}$

Step 5.2.2.3

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

$f'' (- 2) = \frac{- 64 ({(- 2)}^{2} + 48)}{8 {(- 6)}^{3}}$

Step 5.2.2.4

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

$f'' (- 2) = \frac{- 64 ({(- 2)}^{2} + 48)}{8 \cdot - 216}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

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

$f'' (- 2) = \frac{- 64 (4 + 48)}{8 \cdot - 216}$

Step 5.2.3.2

Add $4$ and $48$ .

$f'' (- 2) = \frac{- 64 \cdot 52}{8 \cdot - 216}$

Step 5.2.4

Reduce the expression by cancelling the common factors.

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Step 5.2.4.1

Multiply $8$ by $- 216$ .

$f'' (- 2) = \frac{- 64 \cdot 52}{- 1728}$

Step 5.2.4.2

Multiply $- 64$ by $52$ .

$f'' (- 2) = \frac{- 3328}{- 1728}$

Step 5.2.4.3

Cancel the common factor of $- 3328$ and $- 1728$ .

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Step 5.2.4.3.1

Factor $- 64$ out of $- 3328$ .

$f'' (- 2) = \frac{- 64 \cdot 52}{- 1728}$

Step 5.2.4.3.2

Cancel the common factors.

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Step 5.2.4.3.2.1

Factor $- 64$ out of $- 1728$ .

$f'' (- 2) = \frac{- 64 \cdot 52}{- 64 \cdot 27}$

Step 5.2.4.3.2.2

Cancel the common factor.

$f'' (- 2) = \frac{- 64 \cdot 52}{- 64 \cdot 27}$

Step 5.2.4.3.2.3

Rewrite the expression.

$f'' (- 2) = \frac{52}{27}$

Step 5.2.5

The final answer is $\frac{52}{27}$ .

$\frac{52}{27}$

Step 5.3

The graph is concave up on the interval $(- 4, 0)$ because $f'' (- 2)$ is positive.

Concave up on $(- 4, 0)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(0, 4)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

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

$f'' (2) = \frac{32 (2) ({(2)}^{2} + 48)}{{((2) + 4)}^{3} {((2) - 4)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $32$ by $2$ .

$f'' (2) = \frac{64 (2^{2} + 48)}{{(2 + 4)}^{3} {(2 - 4)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $2$ and $4$ .

$f'' (2) = \frac{64 (2^{2} + 48)}{6^{3} {(2 - 4)}^{3}}$

Step 6.2.2.2

Subtract $4$ from $2$ .

$f'' (2) = \frac{64 (2^{2} + 48)}{6^{3} {(- 2)}^{3}}$

Step 6.2.2.3

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

$f'' (2) = \frac{64 (2^{2} + 48)}{216 {(- 2)}^{3}}$

Step 6.2.2.4

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

$f'' (2) = \frac{64 (2^{2} + 48)}{216 \cdot - 8}$

Step 6.2.3

Simplify the numerator.

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Step 6.2.3.1

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

$f'' (2) = \frac{64 (4 + 48)}{216 \cdot - 8}$

Step 6.2.3.2

Add $4$ and $48$ .

$f'' (2) = \frac{64 \cdot 52}{216 \cdot - 8}$

Step 6.2.4

Reduce the expression by cancelling the common factors.

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Step 6.2.4.1

Multiply $216$ by $- 8$ .

$f'' (2) = \frac{64 \cdot 52}{- 1728}$

Step 6.2.4.2

Multiply $64$ by $52$ .

$f'' (2) = \frac{3328}{- 1728}$

Step 6.2.4.3

Cancel the common factor of $3328$ and $- 1728$ .

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Step 6.2.4.3.1

Factor $64$ out of $3328$ .

$f'' (2) = \frac{64 (52)}{- 1728}$

Step 6.2.4.3.2

Cancel the common factors.

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Step 6.2.4.3.2.1

Factor $64$ out of $- 1728$ .

$f'' (2) = \frac{64 \cdot 52}{64 \cdot - 27}$

Step 6.2.4.3.2.2

Cancel the common factor.

$f'' (2) = \frac{64 \cdot 52}{64 \cdot - 27}$

Step 6.2.4.3.2.3

Rewrite the expression.

$f'' (2) = \frac{52}{- 27}$

Step 6.2.4.4

Move the negative in front of the fraction.

$f'' (2) = - \frac{52}{27}$

Step 6.2.5

The final answer is $- \frac{52}{27}$ .

$- \frac{52}{27}$

Step 6.3

The graph is concave down on the interval $(0, 4)$ because $f'' (2)$ is negative.

Concave down on $(0, 4)$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(4, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

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

$f'' (6) = \frac{32 (6) ({(6)}^{2} + 48)}{{((6) + 4)}^{3} {((6) - 4)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $32$ by $6$ .

$f'' (6) = \frac{192 (6^{2} + 48)}{{(6 + 4)}^{3} {(6 - 4)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Add $6$ and $4$ .

$f'' (6) = \frac{192 (6^{2} + 48)}{10^{3} {(6 - 4)}^{3}}$

Step 7.2.2.2

Subtract $4$ from $6$ .

$f'' (6) = \frac{192 (6^{2} + 48)}{10^{3} \cdot 2^{3}}$

Step 7.2.2.3

Raise $10$ to the power of $3$ .

$f'' (6) = \frac{192 (6^{2} + 48)}{1000 \cdot 2^{3}}$

Step 7.2.2.4

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

$f'' (6) = \frac{192 (6^{2} + 48)}{1000 \cdot 8}$

Step 7.2.3

Simplify the numerator.

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Step 7.2.3.1

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

$f'' (6) = \frac{192 (36 + 48)}{1000 \cdot 8}$

Step 7.2.3.2

Add $36$ and $48$ .

$f'' (6) = \frac{192 \cdot 84}{1000 \cdot 8}$

Step 7.2.4

Reduce the expression by cancelling the common factors.

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Step 7.2.4.1

Multiply $1000$ by $8$ .

$f'' (6) = \frac{192 \cdot 84}{8000}$

Step 7.2.4.2

Multiply $192$ by $84$ .

$f'' (6) = \frac{16128}{8000}$

Step 7.2.4.3

Cancel the common factor of $16128$ and $8000$ .

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Step 7.2.4.3.1

Factor $64$ out of $16128$ .

$f'' (6) = \frac{64 (252)}{8000}$

Step 7.2.4.3.2

Cancel the common factors.

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Step 7.2.4.3.2.1

Factor $64$ out of $8000$ .

$f'' (6) = \frac{64 \cdot 252}{64 \cdot 125}$

Step 7.2.4.3.2.2

Cancel the common factor.

$f'' (6) = \frac{64 \cdot 252}{64 \cdot 125}$

Step 7.2.4.3.2.3

Rewrite the expression.

$f'' (6) = \frac{252}{125}$

Step 7.2.5

The final answer is $\frac{252}{125}$ .

$\frac{252}{125}$

Step 7.3

The graph is concave up on the interval $(4, \infty)$ because $f'' (6)$ is positive.

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 8

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, - 4)$ since $f'' (x)$ is negative

Concave up on $(- 4, 0)$ since $f'' (x)$ is positive

Concave down on $(0, 4)$ since $f'' (x)$ is negative

Concave up on $(4, \infty)$ since $f'' (x)$ is positive

Step 9