Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{4} - 16}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{4} - 16}]$ .

$2 \frac{d}{d x} [\frac{1}{x^{4} - 16}]$

Step 1.1.2

Rewrite $\frac{1}{x^{4} - 16}$ as ${(x^{4} - 16)}^{- 1}$ .

$2 \frac{d}{d x} [{(x^{4} - 16)}^{- 1}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{4} - 16$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{4} - 16$ .

$2 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4} - 16])$

Step 1.2.2

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

$2 (- u^{- 2} \frac{d}{d x} [x^{4} - 16])$

Step 1.2.3

Replace all occurrences of $u$ with $x^{4} - 16$ .

$2 (- {(x^{4} - 16)}^{- 2} \frac{d}{d x} [x^{4} - 16])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $2$ .

$- 2 ({(x^{4} - 16)}^{- 2} \frac{d}{d x} [x^{4} - 16])$

Step 1.3.2

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

$- 2 {(x^{4} - 16)}^{- 2} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16])$

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 2 {(x^{4} - 16)}^{- 2} (4 x^{3} + \frac{d}{d x} [- 16])$

Step 1.3.4

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

$- 2 {(x^{4} - 16)}^{- 2} (4 x^{3} + 0)$

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Add $4 x^{3}$ and $0$ .

$- 2 {(x^{4} - 16)}^{- 2} (4 x^{3})$

Step 1.3.5.2

Multiply $4$ by $- 2$ .

$- 8 {(x^{4} - 16)}^{- 2} x^{3}$

Step 1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 8 \frac{1}{{(x^{4} - 16)}^{2}} x^{3}$

Step 1.5

Combine terms.

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Step 1.5.1

Combine $- 8$ and $\frac{1}{{(x^{4} - 16)}^{2}}$ .

$\frac{- 8}{{(x^{4} - 16)}^{2}} x^{3}$

Step 1.5.2

Move the negative in front of the fraction.

$- \frac{8}{{(x^{4} - 16)}^{2}} x^{3}$

Step 1.5.3

Combine $x^{3}$ and $\frac{8}{{(x^{4} - 16)}^{2}}$ .

$- \frac{x^{3} \cdot 8}{{(x^{4} - 16)}^{2}}$

Step 1.5.4

Move $8$ to the left of $x^{3}$ .

$- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{x^{3}}{{(x^{4} - 16)}^{2}}]$ .

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

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{3}$ and $g (x) = {(x^{4} - 16)}^{2}$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

Multiply the exponents in ${({(x^{4} - 16)}^{2})}^{2}$ .

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $2$ by $2$ .

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

Step 2.3.2

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

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

Step 2.3.3

Move $3$ to the left of ${(x^{4} - 16)}^{2}$ .

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

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{4} - 16$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x^{4} - 16$ .

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

Step 2.4.2

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

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

Step 2.4.3

Replace all occurrences of $u$ with $x^{4} - 16$ .

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

Step 2.5

Differentiate.

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Step 2.5.1

Multiply $2$ by $- 1$ .

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

Step 2.5.2

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

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

Step 2.5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 2.5.4

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

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

Step 2.5.5

Simplify the expression.

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Step 2.5.5.1

Add $4 x^{3}$ and $0$ .

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

Step 2.5.5.2

Move $4$ to the left of $x^{4} - 16$ .

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

Step 2.5.5.3

Multiply $4$ by $- 2$ .

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

Step 2.6

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

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

Step 2.7

Add $3$ and $3$ .

$f'' (x) = - 8 \frac{3 {(x^{4} - 16)}^{2} x^{2} - 8 x^{6} (x^{4} - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.8

Combine $- 8$ and $\frac{3 {(x^{4} - 16)}^{2} x^{2} - 8 x^{6} (x^{4} - 16)}{{(x^{4} - 16)}^{4}}$ .

$f'' (x) = \frac{- 8 (3 {(x^{4} - 16)}^{2} x^{2} - 8 x^{6} (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.9

Move the negative in front of the fraction.

$f'' (x) = - \frac{8 (3 {(x^{4} - 16)}^{2} x^{2} - 8 x^{6} (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$f'' (x) = - \frac{8 (3 {(x^{4} - 16)}^{2} x^{2} - 8 x^{6} x^{4} - 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.2

Apply the distributive property.

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

Step 2.10.3

Simplify the numerator.

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Step 2.10.3.1

Simplify each term.

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Step 2.10.3.1.1

Rewrite ${(x^{4} - 16)}^{2}$ as $(x^{4} - 16) (x^{4} - 16)$ .

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

Step 2.10.3.1.2

Expand $(x^{4} - 16) (x^{4} - 16)$ using the FOIL Method.

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Step 2.10.3.1.2.1

Apply the distributive property.

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

Step 2.10.3.1.2.2

Apply the distributive property.

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

Step 2.10.3.1.2.3

Apply the distributive property.

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

Step 2.10.3.1.3

Simplify and combine like terms.

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Step 2.10.3.1.3.1

Simplify each term.

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Step 2.10.3.1.3.1.1

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

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Step 2.10.3.1.3.1.1.1

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

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

Step 2.10.3.1.3.1.1.2

Add $4$ and $4$ .

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

Step 2.10.3.1.3.1.2

Move $- 16$ to the left of $x^{4}$ .

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

Step 2.10.3.1.3.1.3

Multiply $- 16$ by $- 16$ .

$f'' (x) = - \frac{8 (3 (x^{8} - 16 x^{4} - 16 x^{4} + 256) x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.3.2

Subtract $16 x^{4}$ from $- 16 x^{4}$ .

$f'' (x) = - \frac{8 (3 (x^{8} - 32 x^{4} + 256) x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.4

Apply the distributive property.

$f'' (x) = - \frac{8 ((3 x^{8} + 3 (- 32 x^{4}) + 3 \cdot 256) x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.5

Simplify.

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Step 2.10.3.1.5.1

Multiply $- 32$ by $3$ .

$f'' (x) = - \frac{8 ((3 x^{8} - 96 x^{4} + 3 \cdot 256) x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.5.2

Multiply $3$ by $256$ .

$f'' (x) = - \frac{8 ((3 x^{8} - 96 x^{4} + 768) x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.6

Apply the distributive property.

$f'' (x) = - \frac{8 (3 x^{8} x^{2} - 96 x^{4} x^{2} + 768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.7

Simplify.

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Step 2.10.3.1.7.1

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

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Step 2.10.3.1.7.1.1

Move $x^{2}$ .

$f'' (x) = - \frac{8 (3 (x^{2} x^{8}) - 96 x^{4} x^{2} + 768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.7.1.2

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

$f'' (x) = - \frac{8 (3 x^{2 + 8} - 96 x^{4} x^{2} + 768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.7.1.3

Add $2$ and $8$ .

$f'' (x) = - \frac{8 (3 x^{10} - 96 x^{4} x^{2} + 768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.7.2

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

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Step 2.10.3.1.7.2.1

Move $x^{2}$ .

$f'' (x) = - \frac{8 (3 x^{10} - 96 (x^{2} x^{4}) + 768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.7.2.2

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

$f'' (x) = - \frac{8 (3 x^{10} - 96 x^{2 + 4} + 768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.7.2.3

Add $2$ and $4$ .

$f'' (x) = - \frac{8 (3 x^{10} - 96 x^{6} + 768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.8

Apply the distributive property.

$f'' (x) = - \frac{8 (3 x^{10}) + 8 (- 96 x^{6}) + 8 (768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.9

Simplify.

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Step 2.10.3.1.9.1

Multiply $3$ by $8$ .

$f'' (x) = - \frac{24 x^{10} + 8 (- 96 x^{6}) + 8 (768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.9.2

Multiply $- 96$ by $8$ .

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 8 (768 x^{2}) + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.9.3

Multiply $768$ by $8$ .

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 6144 x^{2} + 8 (- 8 x^{6} x^{4}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.10

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

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Step 2.10.3.1.10.1

Move $x^{4}$ .

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 6144 x^{2} + 8 (- 8 (x^{4} x^{6})) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.10.2

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

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 6144 x^{2} + 8 (- 8 x^{4 + 6}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.10.3

Add $4$ and $6$ .

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 6144 x^{2} + 8 (- 8 x^{10}) + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.11

Multiply $- 8$ by $8$ .

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 6144 x^{2} - 64 x^{10} + 8 (- 8 x^{6} \cdot - 16)}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.12

Multiply $- 16$ by $- 8$ .

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 6144 x^{2} - 64 x^{10} + 8 (128 x^{6})}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.1.13

Multiply $128$ by $8$ .

$f'' (x) = - \frac{24 x^{10} - 768 x^{6} + 6144 x^{2} - 64 x^{10} + 1024 x^{6}}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.2

Subtract $64 x^{10}$ from $24 x^{10}$ .

$f'' (x) = - \frac{- 40 x^{10} - 768 x^{6} + 6144 x^{2} + 1024 x^{6}}{{(x^{4} - 16)}^{4}}$

Step 2.10.3.3

Add $- 768 x^{6}$ and $1024 x^{6}$ .

$f'' (x) = - \frac{- 40 x^{10} + 256 x^{6} + 6144 x^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.10.4

Simplify the numerator.

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Step 2.10.4.1

Factor $8 x^{2}$ out of $- 40 x^{10} + 256 x^{6} + 6144 x^{2}$ .

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Step 2.10.4.1.1

Factor $8 x^{2}$ out of $- 40 x^{10}$ .

$f'' (x) = - \frac{8 x^{2} (- 5 x^{8}) + 256 x^{6} + 6144 x^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.1.2

Factor $8 x^{2}$ out of $256 x^{6}$ .

$f'' (x) = - \frac{8 x^{2} (- 5 x^{8}) + 8 x^{2} (32 x^{4}) + 6144 x^{2}}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.1.3

Factor $8 x^{2}$ out of $6144 x^{2}$ .

$f'' (x) = - \frac{8 x^{2} (- 5 x^{8}) + 8 x^{2} (32 x^{4}) + 8 x^{2} (768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.1.4

Factor $8 x^{2}$ out of $8 x^{2} (- 5 x^{8}) + 8 x^{2} (32 x^{4})$ .

$f'' (x) = - \frac{8 x^{2} (- 5 x^{8} + 32 x^{4}) + 8 x^{2} (768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.1.5

Factor $8 x^{2}$ out of $8 x^{2} (- 5 x^{8} + 32 x^{4}) + 8 x^{2} (768)$ .

$f'' (x) = - \frac{8 x^{2} (- 5 x^{8} + 32 x^{4} + 768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$f'' (x) = - \frac{8 x^{2} (- 5 {(x^{4})}^{2} + 32 x^{4} + 768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.3

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

$f'' (x) = - \frac{8 x^{2} (- 5 u^{2} + 32 u + 768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.4

Factor by grouping.

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Step 2.10.4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 5 \cdot 768 = - 3840$ and whose sum is $b = 32$ .

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Step 2.10.4.4.1.1

Factor $32$ out of $32 u$ .

$f'' (x) = - \frac{8 x^{2} (- 5 u^{2} + 32 (u) + 768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.4.1.2

Rewrite $32$ as $- 48$ plus $80$

$f'' (x) = - \frac{8 x^{2} (- 5 u^{2} + (- 48 + 80) u + 768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.4.1.3

Apply the distributive property.

$f'' (x) = - \frac{8 x^{2} (- 5 u^{2} - 48 u + 80 u + 768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.4.2

Factor out the greatest common factor from each group.

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Step 2.10.4.4.2.1

Group the first two terms and the last two terms.

$f'' (x) = - \frac{8 x^{2} ((- 5 u^{2} - 48 u) + 80 u + 768)}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.4.2.2

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = - \frac{8 x^{2} (u (- 5 u - 48) - 16 (- 5 u - 48))}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.4.3

Factor the polynomial by factoring out the greatest common factor, $- 5 u - 48$ .

$f'' (x) = - \frac{8 x^{2} ((- 5 u - 48) (u - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.5

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

$f'' (x) = - \frac{8 x^{2} ((- 5 x^{4} - 48) (x^{4} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.6

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

$f'' (x) = - \frac{8 x^{2} ((- 5 x^{4} - 48) ({(x^{2})}^{2} - 16))}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.7

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

$f'' (x) = - \frac{8 x^{2} ((- 5 x^{4} - 48) ({(x^{2})}^{2} - 4^{2}))}{{(x^{4} - 16)}^{4}}$

Step 2.10.4.8

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

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

Step 2.10.4.9

Factor.

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

Step 2.10.5

Simplify the denominator.

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Step 2.10.5.1

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

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

Step 2.10.5.2

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

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

Step 2.10.5.3

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

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

Step 2.10.5.4

Simplify.

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Step 2.10.5.4.1

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

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

Step 2.10.5.4.2

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

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

Step 2.10.5.5

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

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

Step 2.10.5.6

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

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Step 2.10.5.6.1

Apply the distributive property.

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

Step 2.10.5.6.2

Apply the distributive property.

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

Step 2.10.5.6.3

Apply the distributive property.

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

Step 2.10.5.7

Simplify each term.

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Step 2.10.5.7.1

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

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Step 2.10.5.7.1.1

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

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Step 2.10.5.7.1.1.1

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

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

Step 2.10.5.7.1.1.2

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

$f'' (x) = - \frac{8 x^{2} (- 5 x^{4} - 48) (x^{2} + 4) (x + 2) (x - 2)}{{(x^{2 + 1} + x^{2} \cdot 2 + 4 x + 4 \cdot 2)}^{4} {(x - 2)}^{4}}$

Step 2.10.5.7.1.2

Add $2$ and $1$ .

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

Step 2.10.5.7.2

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

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

Step 2.10.5.7.3

Multiply $4$ by $2$ .

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

Step 2.10.5.8

Factor out the greatest common factor from each group.

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Step 2.10.5.8.1

Group the first two terms and the last two terms.

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

Step 2.10.5.8.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.10.5.9

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

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

Step 2.10.5.10

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

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

Step 2.10.6

Cancel the common factor of $x^{2} + 4$ and ${(x^{2} + 4)}^{4}$ .

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Step 2.10.6.1

Factor $x^{2} + 4$ out of $8 x^{2} (- 5 x^{4} - 48) (x^{2} + 4) (x + 2) (x - 2)$ .

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

Step 2.10.6.2

Cancel the common factors.

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Step 2.10.6.2.1

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

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

Step 2.10.6.2.2

Cancel the common factor.

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

Step 2.10.6.2.3

Rewrite the expression.

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

Step 2.10.7

Cancel the common factor of $x + 2$ and ${(x + 2)}^{4}$ .

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Step 2.10.7.1

Factor $x + 2$ out of $(8 x^{2} (- 5 x^{4} - 48) (x + 2)) (x - 2)$ .

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

Step 2.10.7.2

Cancel the common factors.

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Step 2.10.7.2.1

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

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

Step 2.10.7.2.2

Cancel the common factor.

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

Step 2.10.7.2.3

Rewrite the expression.

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

Step 2.10.8

Cancel the common factor of $x - 2$ and ${(x - 2)}^{4}$ .

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Step 2.10.8.1

Factor $x - 2$ out of $(8 x^{2} (- 5 x^{4} - 48)) (x - 2)$ .

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

Step 2.10.8.2

Cancel the common factors.

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Step 2.10.8.2.1

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

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

Step 2.10.8.2.2

Cancel the common factor.

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

Step 2.10.8.2.3

Rewrite the expression.

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

Step 2.10.9

Factor $- 1$ out of $- 5 x^{4}$ .

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

Step 2.10.10

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

$f'' (x) = - \frac{8 x^{2} (- (5 x^{4}) - 1 \cdot 48)}{{(x + 2)}^{3} {(x^{2} + 4)}^{3} {(x - 2)}^{3}}$

Step 2.10.11

Factor $- 1$ out of $- (5 x^{4}) - 1 (48)$ .

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

Step 2.10.12

Rewrite $- (5 x^{4} + 48)$ as $- 1 (5 x^{4} + 48)$ .

$f'' (x) = - \frac{8 x^{2} (- 1 (5 x^{4} + 48))}{{(x + 2)}^{3} {(x^{2} + 4)}^{3} {(x - 2)}^{3}}$

Step 2.10.13

Move the negative in front of the fraction.

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

Step 2.10.14

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{(8 x^{2}) (5 x^{4} + 48)}{{(x + 2)}^{3} {(x^{2} + 4)}^{3} {(x - 2)}^{3}})$

Step 2.10.15

Multiply $\frac{(8 x^{2}) (5 x^{4} + 48)}{{(x + 2)}^{3} {(x^{2} + 4)}^{3} {(x - 2)}^{3}}$ by $1$ .

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

Step 2.10.16

Reorder factors in $\frac{(8 x^{2}) (5 x^{4} + 48)}{{(x + 2)}^{3} {(x^{2} + 4)}^{3} {(x - 2)}^{3}}$ .

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

Step 3

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

$- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{4} - 16}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{4} - 16}]$ .

$2 \frac{d}{d x} [\frac{1}{x^{4} - 16}]$

Step 4.1.1.2

Rewrite $\frac{1}{x^{4} - 16}$ as ${(x^{4} - 16)}^{- 1}$ .

$2 \frac{d}{d x} [{(x^{4} - 16)}^{- 1}]$

Step 4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{4} - 16$ .

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Step 4.1.2.1

To apply the Chain Rule, set $u$ as $x^{4} - 16$ .

$2 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4} - 16])$

Step 4.1.2.2

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

$2 (- u^{- 2} \frac{d}{d x} [x^{4} - 16])$

Step 4.1.2.3

Replace all occurrences of $u$ with $x^{4} - 16$ .

$2 (- {(x^{4} - 16)}^{- 2} \frac{d}{d x} [x^{4} - 16])$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

Multiply $- 1$ by $2$ .

$- 2 ({(x^{4} - 16)}^{- 2} \frac{d}{d x} [x^{4} - 16])$

Step 4.1.3.2

By the Sum Rule, the derivative of $x^{4} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16]$ .

$- 2 {(x^{4} - 16)}^{- 2} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 16])$

Step 4.1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 2 {(x^{4} - 16)}^{- 2} (4 x^{3} + \frac{d}{d x} [- 16])$

Step 4.1.3.4

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

$- 2 {(x^{4} - 16)}^{- 2} (4 x^{3} + 0)$

Step 4.1.3.5

Simplify the expression.

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Step 4.1.3.5.1

Add $4 x^{3}$ and $0$ .

$- 2 {(x^{4} - 16)}^{- 2} (4 x^{3})$

Step 4.1.3.5.2

Multiply $4$ by $- 2$ .

$- 8 {(x^{4} - 16)}^{- 2} x^{3}$

Step 4.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 8 \frac{1}{{(x^{4} - 16)}^{2}} x^{3}$

Step 4.1.5

Combine terms.

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Step 4.1.5.1

Combine $- 8$ and $\frac{1}{{(x^{4} - 16)}^{2}}$ .

$\frac{- 8}{{(x^{4} - 16)}^{2}} x^{3}$

Step 4.1.5.2

Move the negative in front of the fraction.

$- \frac{8}{{(x^{4} - 16)}^{2}} x^{3}$

Step 4.1.5.3

Combine $x^{3}$ and $\frac{8}{{(x^{4} - 16)}^{2}}$ .

$- \frac{x^{3} \cdot 8}{{(x^{4} - 16)}^{2}}$

Step 4.1.5.4

Move $8$ to the left of $x^{3}$ .

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$ .

$- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}} = 0$

Step 5.2

Set the numerator equal to zero.

$8 x^{3} = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Divide each term in $8 x^{3} = 0$ by $8$ and simplify.

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Step 5.3.1.1

Divide each term in $8 x^{3} = 0$ by $8$ .

$\frac{8 x^{3}}{8} = \frac{0}{8}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $8$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

$\frac{8 x^{3}}{8} = \frac{0}{8}$

Step 5.3.1.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{0}{8}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $8$ .

$x^{3} = 0$

Step 5.3.2

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

$x = \sqrt[3]{0}$

Step 5.3.3

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

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Step 5.3.3.1

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

$x = \sqrt[3]{0^{3}}$

Step 5.3.3.2

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

$x = 0$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 6.2

Solve for $x$ .

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Step 6.2.1

Factor the left side of the equation.

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Step 6.2.1.1

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

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Simplify.

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Step 6.2.1.4.1

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

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

Step 6.2.1.4.2

Factor.

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Step 6.2.1.4.2.1

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 $b = 2$ .

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

Step 6.2.1.4.2.2

Remove unnecessary parentheses.

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

Step 6.2.1.5

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

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

Step 6.2.1.6

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

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

Step 6.2.2

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

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

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

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

Step 6.2.3

Set ${(x^{2} + 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 6.2.3.1

Set ${(x^{2} + 4)}^{2}$ equal to $0$ .

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

Step 6.2.3.2

Solve ${(x^{2} + 4)}^{2} = 0$ for $x$ .

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Step 6.2.3.2.1

Set the $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Step 6.2.3.2.2

Solve for $x$ .

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Step 6.2.3.2.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 6.2.3.2.2.2

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

$x = \pm \sqrt{- 4}$

Step 6.2.3.2.2.3

Simplify $\pm \sqrt{- 4}$ .

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Step 6.2.3.2.2.3.1

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

$x = \pm \sqrt{- 1 (4)}$

Step 6.2.3.2.2.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 6.2.3.2.2.3.3

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

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

Step 6.2.3.2.2.3.4

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

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

Step 6.2.3.2.2.3.5

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

$x = \pm i \cdot 2$

Step 6.2.3.2.2.3.6

Move $2$ to the left of $i$ .

$x = \pm 2 i$

Step 6.2.3.2.2.4

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

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Step 6.2.3.2.2.4.1

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

$x = 2 i$

Step 6.2.3.2.2.4.2

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

$x = - 2 i$

Step 6.2.3.2.2.4.3

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

$x = 2 i, - 2 i$

Step 6.2.4

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

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Step 6.2.4.1

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

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

Step 6.2.4.2

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

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Step 6.2.4.2.1

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

$x + 2 = 0$

Step 6.2.4.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6.2.5

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

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Step 6.2.5.1

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

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

Step 6.2.5.2

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

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Step 6.2.5.2.1

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

$x - 2 = 0$

Step 6.2.5.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6.2.6

The final solution is all the values that make ${(x^{2} + 4)}^{2} {(x + 2)}^{2} {(x - 2)}^{2} = 0$ true.

$x = 2 i, - 2 i, - 2, 2$

Step 6.3

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

$x = - 2, x = 2$

Step 7

Critical points to evaluate.

$x = 0$

Step 8

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.

$\frac{8 {(0)}^{2} (5 {(0)}^{4} + 48)}{{((0) + 2)}^{3} {({(0)}^{2} + 4)}^{3} {((0) - 2)}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{8 \cdot 0^{2} (5 \cdot 0 + 48)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.2

Multiply $5$ by $0$ .

$\frac{8 \cdot 0^{2} (0 + 48)}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.3

Add $0$ and $48$ .

$\frac{8 \cdot 0^{2} \cdot 48}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.4

Multiply $48$ by $8$ .

$\frac{384 \cdot 0^{2}}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.1.5

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

$\frac{384 \cdot 0}{{(0 + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Rewrite $0$ as $- 1 (0)$ .

$\frac{384 \cdot 0}{{(- 1 (0) + 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{384 \cdot 0}{{(- 1 \cdot 0 - 1 \cdot - 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 2$ .

$\frac{384 \cdot 0}{{(- 1 \cdot (0 - 2))}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.4

Apply the product rule to $- 1 \cdot (0 - 2)$ .

$\frac{384 \cdot 0}{{(- 1)}^{3} \cdot {(0 - 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.5

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

$\frac{384 \cdot 0}{- 1 \cdot {(0 - 2)}^{3} {(0^{2} + 4)}^{3} {(0 - 2)}^{3}}$

Step 9.2.6

Multiply ${(0 - 2)}^{3}$ by ${(0 - 2)}^{3}$ by adding the exponents.

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Step 9.2.6.1

Move ${(0 - 2)}^{3}$ .

$\frac{384 \cdot 0}{- 1 \cdot ({(0 - 2)}^{3} {(0 - 2)}^{3}) {(0^{2} + 4)}^{3}}$

Step 9.2.6.2

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

$\frac{384 \cdot 0}{- 1 \cdot {(0 - 2)}^{3 + 3} {(0^{2} + 4)}^{3}}$

Step 9.2.6.3

Add $3$ and $3$ .

$\frac{384 \cdot 0}{- 1 \cdot {(0 - 2)}^{6} {(0^{2} + 4)}^{3}}$

Step 9.3

Multiply $384$ by $0$ .

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

Step 9.4

Simplify the denominator.

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Step 9.4.1

Subtract $2$ from $0$ .

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

Step 9.4.2

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

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

Step 9.4.3

Add $0$ and $4$ .

$\frac{0}{- 1 \cdot {(- 2)}^{6} \cdot 4^{3}}$

Step 9.4.4

Combine exponents.

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Step 9.4.4.1

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{0}{- 1 \cdot {(- 1 (2))}^{6} \cdot 4^{3}}$

Step 9.4.4.2

Apply the product rule to $- 1 (2)$ .

$\frac{0}{- 1 \cdot ({(- 1)}^{6} \cdot 2^{6}) \cdot 4^{3}}$

Step 9.4.4.3

Raise $- 1$ to the power of $6$ .

$\frac{0}{- 1 \cdot (1 \cdot 2^{6}) \cdot 4^{3}}$

Step 9.4.4.4

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

$\frac{0}{- 1 \cdot 2^{6} \cdot 4^{3}}$

Step 9.4.4.5

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

$\frac{0}{- 1 \cdot ({(2^{2})}^{3} \cdot 2^{6})}$

Step 9.4.4.6

Multiply the exponents in ${(2^{2})}^{3}$ .

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Step 9.4.4.6.1

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

$\frac{0}{- 1 \cdot (2^{2 \cdot 3} \cdot 2^{6})}$

Step 9.4.4.6.2

Multiply $2$ by $3$ .

$\frac{0}{- 1 \cdot (2^{6} \cdot 2^{6})}$

Step 9.4.4.7

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

$\frac{0}{- 1 \cdot 2^{6 + 6}}$

Step 9.4.4.8

Add $6$ and $6$ .

$\frac{0}{- 1 \cdot 2^{12}}$

Step 9.4.5

Raise $2$ to the power of $12$ .

$\frac{0}{- 1 \cdot 4096}$

Step 9.5

Simplify the expression.

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Step 9.5.1

Multiply $- 1$ by $4096$ .

$\frac{0}{- 4096}$

Step 9.5.2

Divide $0$ by $- 4096$ .

$0$

Step 10

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

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Step 10.1

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

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

Step 10.2

Substitute any number, such as $- 4$ , from the interval $(- \infty, 0)$ in the first derivative $- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Raise $- 4$ to the power of $3$ .

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

Step 10.2.2.2

Simplify the denominator.

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Step 10.2.2.2.1

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

$f' (- 4) = - \frac{8 \cdot - 64}{{(256 - 16)}^{2}}$

Step 10.2.2.2.2

Subtract $16$ from $256$ .

$f' (- 4) = - \frac{8 \cdot - 64}{240^{2}}$

Step 10.2.2.2.3

Raise $240$ to the power of $2$ .

$f' (- 4) = - \frac{8 \cdot - 64}{57600}$

Step 10.2.2.3

Reduce the expression by cancelling the common factors.

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Step 10.2.2.3.1

Multiply $8$ by $- 64$ .

$f' (- 4) = - \frac{- 512}{57600}$

Step 10.2.2.3.2

Cancel the common factor of $- 512$ and $57600$ .

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Step 10.2.2.3.2.1

Factor $256$ out of $- 512$ .

$f' (- 4) = - \frac{256 (- 2)}{57600}$

Step 10.2.2.3.2.2

Cancel the common factors.

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Step 10.2.2.3.2.2.1

Factor $256$ out of $57600$ .

$f' (- 4) = - \frac{256 \cdot - 2}{256 \cdot 225}$

Step 10.2.2.3.2.2.2

Cancel the common factor.

$f' (- 4) = - \frac{256 \cdot - 2}{256 \cdot 225}$

Step 10.2.2.3.2.2.3

Rewrite the expression.

$f' (- 4) = - \frac{- 2}{225}$

Step 10.2.2.3.3

Move the negative in front of the fraction.

$f' (- 4) = \frac{2}{225}$

Step 10.2.2.4

The final answer is $\frac{2}{225}$ .

$\frac{2}{225}$

Step 10.3

Substitute any number, such as $1$ , from the interval $(0, \infty)$ in the first derivative $- \frac{8 x^{3}}{{(x^{4} - 16)}^{2}}$ to check if the result is negative or positive.

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Step 10.3.1

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

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

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

One to any power is one.

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

Step 10.3.2.2

Simplify the denominator.

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Step 10.3.2.2.1

One to any power is one.

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

Step 10.3.2.2.2

Subtract $16$ from $1$ .

$f' (1) = - \frac{8 \cdot 1}{{(- 15)}^{2}}$

Step 10.3.2.2.3

Raise $- 15$ to the power of $2$ .

$f' (1) = - \frac{8 \cdot 1}{225}$

Step 10.3.2.3

Multiply $8$ by $1$ .

$f' (1) = - \frac{8}{225}$

Step 10.3.2.4

The final answer is $- \frac{8}{225}$ .

$- \frac{8}{225}$

Step 10.4

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 11