Calculus Examples

$\frac{8}{x^{2} - 4 x}$

Step 1

Write $\frac{8}{x^{2} - 4 x}$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x^{2} - 4 x}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x^{2} - 4 x}]$ .

$8 \frac{d}{d x} [\frac{1}{x^{2} - 4 x}]$

Step 2.1.2

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

$8 \frac{d}{d x} [{(x^{2} - 4 x)}^{- 1}]$

Step 2.2

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

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Step 2.2.1

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

$8 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 4 x])$

Step 2.2.2

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

$8 (- u^{- 2} \frac{d}{d x} [x^{2} - 4 x])$

Step 2.2.3

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

$8 (- {(x^{2} - 4 x)}^{- 2} \frac{d}{d x} [x^{2} - 4 x])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $8$ .

$- 8 ({(x^{2} - 4 x)}^{- 2} \frac{d}{d x} [x^{2} - 4 x])$

Step 2.3.2

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

$- 8 {(x^{2} - 4 x)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x])$

Step 2.3.3

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

$- 8 {(x^{2} - 4 x)}^{- 2} (2 x + \frac{d}{d x} [- 4 x])$

Step 2.3.4

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

$- 8 {(x^{2} - 4 x)}^{- 2} (2 x - 4 \frac{d}{d x} [x])$

Step 2.3.5

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

$- 8 {(x^{2} - 4 x)}^{- 2} (2 x - 4 \cdot 1)$

Step 2.3.6

Multiply $- 4$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

Combine terms.

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Step 2.4.2.1

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

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

Step 2.4.2.2

Move the negative in front of the fraction.

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

Step 2.4.3

Reorder the factors of $- \frac{8}{{(x^{2} - 4 x)}^{2}} (2 x - 4)$ .

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

Step 2.4.4

Apply the distributive property.

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

Step 2.4.5

Multiply $2$ by $- 1$ .

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

Step 2.4.6

Multiply $- 1$ by $- 4$ .

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

Step 2.4.7

Simplify the denominator.

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Step 2.4.7.1

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

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Step 2.4.7.1.1

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

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

Step 2.4.7.1.2

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

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

Step 2.4.7.1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

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

Step 2.4.7.2

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

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

Step 2.4.8

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

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

Step 2.4.9

Simplify the numerator.

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Step 2.4.9.1

Factor $2$ out of $- 2 x + 4$ .

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Step 2.4.9.1.1

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

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

Step 2.4.9.1.2

Factor $2$ out of $4$ .

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

Step 2.4.9.1.3

Factor $2$ out of $2 (- x) + 2 (2)$ .

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

Step 2.4.9.2

Multiply $8$ by $2$ .

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

Step 2.4.10

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

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

Step 2.4.11

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

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

Step 2.4.12

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 2.4.13

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

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

Step 2.4.14

Move the negative in front of the fraction.

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

Step 3

Find the second derivative of the function.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = - 16 \frac{x^{2} {(x - 4)}^{2} (1 + 0) - (x - 2) \frac{d}{d x} (x^{2} {(x - 4)}^{2})}{{(x^{2} {(x - 4)}^{2})}^{2}}$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Add $1$ and $0$ .

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

Step 3.3.4.2

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

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

Step 3.4

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

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

Step 3.5

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 3.5.1

To apply the Chain Rule, set $u$ as $x - 4$ .

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

Step 3.5.2

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

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

Step 3.5.3

Replace all occurrences of $u$ with $x - 4$ .

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

Step 3.6

Differentiate.

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

$f'' (x) = - 16 \frac{x^{2} {(x - 4)}^{2} - (x - 2) (x^{2} (2 (x - 4) (1 + 0)) + {(x - 4)}^{2} \frac{d}{d x} (x^{2}))}{{(x^{2} {(x - 4)}^{2})}^{2}}$

Step 3.6.4

Simplify the expression.

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Step 3.6.4.1

Add $1$ and $0$ .

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

Step 3.6.4.2

Multiply $2$ by $1$ .

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

Step 3.6.5

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

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

Step 3.6.6

Combine fractions.

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Step 3.6.6.1

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

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

Step 3.6.6.2

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

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

Step 3.6.6.3

Move the negative in front of the fraction.

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

Step 3.7

Simplify.

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Step 3.7.1

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

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

Step 3.7.2

Apply the distributive property.

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

Step 3.7.3

Apply the distributive property.

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

Step 3.7.4

Apply the distributive property.

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

Step 3.7.5

Apply the distributive property.

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

Step 3.7.6

Simplify the numerator.

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Step 3.7.6.1

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

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Step 3.7.6.1.1

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

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

Step 3.7.6.1.2

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

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

Step 3.7.6.1.3

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

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

Step 3.7.6.2

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

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

Step 3.7.6.3

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

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Step 3.7.6.3.1

Apply the distributive property.

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

Step 3.7.6.3.2

Apply the distributive property.

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

Step 3.7.6.3.3

Apply the distributive property.

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

Step 3.7.6.4

Simplify and combine like terms.

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Step 3.7.6.4.1

Simplify each term.

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Step 3.7.6.4.1.1

Multiply $x$ by $x$ .

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

Step 3.7.6.4.1.2

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

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

Step 3.7.6.4.1.3

Multiply $- 4$ by $- 4$ .

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

Step 3.7.6.4.2

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

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

Step 3.7.6.5

Apply the distributive property.

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

Step 3.7.6.6

Simplify.

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Step 3.7.6.6.1

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

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Step 3.7.6.6.1.1

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

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

Step 3.7.6.6.1.2

Add $2$ and $2$ .

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

Step 3.7.6.6.2

Rewrite using the commutative property of multiplication.

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

Step 3.7.6.6.3

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

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

Step 3.7.6.7

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

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Step 3.7.6.7.1

Move $x$ .

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

Step 3.7.6.7.2

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

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Step 3.7.6.7.2.1

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

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

Step 3.7.6.7.2.2

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

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

Step 3.7.6.7.3

Add $1$ and $2$ .

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

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

Step 3.7.6.8

Multiply $- 1$ by $- 2$ .

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

Step 3.7.6.9

Simplify each term.

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Step 3.7.6.9.1

Rewrite using the commutative property of multiplication.

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

Step 3.7.6.9.2

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

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Step 3.7.6.9.2.1

Move $x$ .

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

Step 3.7.6.9.2.2

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

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Step 3.7.6.9.2.2.1

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

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

Step 3.7.6.9.2.2.2

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

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

Step 3.7.6.9.2.3

Add $1$ and $2$ .

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

Step 3.7.6.9.3

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

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

Step 3.7.6.9.4

Multiply $- 4$ by $2$ .

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

Step 3.7.6.9.5

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

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

Step 3.7.6.9.6

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

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Step 3.7.6.9.6.1

Apply the distributive property.

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

Step 3.7.6.9.6.2

Apply the distributive property.

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

Step 3.7.6.9.6.3

Apply the distributive property.

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

Step 3.7.6.9.7

Simplify and combine like terms.

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Step 3.7.6.9.7.1

Simplify each term.

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Step 3.7.6.9.7.1.1

Multiply $x$ by $x$ .

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

Step 3.7.6.9.7.1.2

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

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

Step 3.7.6.9.7.1.3

Multiply $- 4$ by $- 4$ .

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

Step 3.7.6.9.7.2

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

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

Step 3.7.6.9.8

Apply the distributive property.

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

Step 3.7.6.9.9

Simplify.

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Step 3.7.6.9.9.1

Multiply $- 8$ by $2$ .

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

Step 3.7.6.9.9.2

Multiply $2$ by $16$ .

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

Step 3.7.6.9.10

Apply the distributive property.

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

Step 3.7.6.9.11

Simplify.

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Step 3.7.6.9.11.1

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

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Step 3.7.6.9.11.1.1

Move $x$ .

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

Step 3.7.6.9.11.1.2

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

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Step 3.7.6.9.11.1.2.1

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

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

Step 3.7.6.9.11.1.2.2

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

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

Step 3.7.6.9.11.1.3

Add $1$ and $2$ .

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

Step 3.7.6.9.11.2

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

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Step 3.7.6.9.11.2.1

Move $x$ .

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

Step 3.7.6.9.11.2.2

Multiply $x$ by $x$ .

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

Step 3.7.6.10

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

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

Step 3.7.6.11

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

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} + (- x + 2) (4 x^{3} - 24 x^{2} + 32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.12

Expand $(- x + 2) (4 x^{3} - 24 x^{2} + 32 x)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - x (4 x^{3}) - x (- 24 x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13

Simplify each term.

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Step 3.7.6.13.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 1 \cdot (4 x \cdot x^{3}) - x (- 24 x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.2

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

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Step 3.7.6.13.2.1

Move $x^{3}$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 1 \cdot (4 (x^{3} x)) - x (- 24 x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.2.2

Multiply $x^{3}$ by $x$ .

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Step 3.7.6.13.2.2.1

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

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 1 \cdot (4 (x^{3} x)) - x (- 24 x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.2.2.2

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

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 1 \cdot (4 x^{3 + 1}) - x (- 24 x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.2.3

Add $3$ and $1$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 1 \cdot (4 x^{4}) - x (- 24 x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.3

Multiply $- 1$ by $4$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} - x (- 24 x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.4

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} - 1 \cdot (- 24 x \cdot x^{2}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.5

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

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Step 3.7.6.13.5.1

Move $x^{2}$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} - 1 \cdot (- 24 (x^{2} x)) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.5.2

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

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Step 3.7.6.13.5.2.1

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

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} - 1 \cdot (- 24 (x^{2} x)) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.5.2.2

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

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} - 1 \cdot (- 24 x^{2 + 1}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.5.3

Add $2$ and $1$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} - 1 \cdot (- 24 x^{3}) - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.6

Multiply $- 1$ by $- 24$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - x (32 x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.7

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - 1 \cdot (32 x \cdot x) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.8

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

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Step 3.7.6.13.8.1

Move $x$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - 1 \cdot (32 (x \cdot x)) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.8.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - 1 \cdot (32 x^{2}) + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.9

Multiply $- 1$ by $32$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - 32 x^{2} + 2 (4 x^{3}) + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.10

Multiply $4$ by $2$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - 32 x^{2} + 8 x^{3} + 2 (- 24 x^{2}) + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.11

Multiply $- 24$ by $2$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - 32 x^{2} + 8 x^{3} - 48 x^{2} + 2 (32 x))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.13.12

Multiply $32$ by $2$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 24 x^{3} - 32 x^{2} + 8 x^{3} - 48 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.14

Add $24 x^{3}$ and $8 x^{3}$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 32 x^{3} - 32 x^{2} - 48 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.15

Subtract $48 x^{2}$ from $- 32 x^{2}$ .

$f'' (x) = - \frac{16 (x^{4} - 8 x^{3} + 16 x^{2} - 4 x^{4} + 32 x^{3} - 80 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.16

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

$f'' (x) = - \frac{16 (- 3 x^{4} - 8 x^{3} + 16 x^{2} + 32 x^{3} - 80 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.17

Add $- 8 x^{3}$ and $32 x^{3}$ .

$f'' (x) = - \frac{16 (- 3 x^{4} + 24 x^{3} + 16 x^{2} - 80 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.18

Subtract $80 x^{2}$ from $16 x^{2}$ .

$f'' (x) = - \frac{16 (- 3 x^{4} + 24 x^{3} - 64 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19

Rewrite $- 3 x^{4} + 24 x^{3} - 64 x^{2} + 64 x$ in a factored form.

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Step 3.7.6.19.1

Factor $x$ out of $- 3 x^{4} + 24 x^{3} - 64 x^{2} + 64 x$ .

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Step 3.7.6.19.1.1

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

$f'' (x) = - \frac{16 (x (- 3 x^{3}) + 24 x^{3} - 64 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19.1.2

Factor $x$ out of $24 x^{3}$ .

$f'' (x) = - \frac{16 (x (- 3 x^{3}) + x (24 x^{2}) - 64 x^{2} + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19.1.3

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

$f'' (x) = - \frac{16 (x (- 3 x^{3}) + x (24 x^{2}) + x (- 64 x) + 64 x)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19.1.4

Factor $x$ out of $64 x$ .

$f'' (x) = - \frac{16 (x (- 3 x^{3}) + x (24 x^{2}) + x (- 64 x) + x \cdot 64)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19.1.5

Factor $x$ out of $x (- 3 x^{3}) + x (24 x^{2})$ .

$f'' (x) = - \frac{16 (x (- 3 x^{3} + 24 x^{2}) + x (- 64 x) + x \cdot 64)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19.1.6

Factor $x$ out of $x (- 3 x^{3} + 24 x^{2}) + x (- 64 x)$ .

$f'' (x) = - \frac{16 (x (- 3 x^{3} + 24 x^{2} - 64 x) + x \cdot 64)}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19.1.7

Factor $x$ out of $x (- 3 x^{3} + 24 x^{2} - 64 x) + x \cdot 64$ .

$f'' (x) = - \frac{16 (x (- 3 x^{3} + 24 x^{2} - 64 x + 64))}{{(x^{2})}^{2} {({(x - 4)}^{2})}^{2}}$

Step 3.7.6.19.2

Factor $- 3 x^{3} + 24 x^{2} - 64 x + 64$ using the rational roots test.

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Step 3.7.6.19.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 64, \pm 2, \pm 32, \pm 4, \pm 16, \pm 8$

$q = \pm 1, \pm 3$

Step 3.7.6.19.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.\overline{3}, \pm 64, \pm 21.\overline{3}, \pm 2, \pm 0.\overline{6}, \pm 32, \pm 10.\overline{6}, \pm 4, \pm 1.\overline{3}, \pm 16, \pm 5.\overline{3}, \pm 8, \pm 2.\overline{6}$

Step 3.7.6.19.2.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 3.7.6.19.2.3.1

Substitute $4$ into the polynomial.

$- 3 \cdot 4^{3} + 24 \cdot 4^{2} - 64 \cdot 4 + 64$

Step 3.7.6.19.2.3.2

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

$- 3 \cdot 64 + 24 \cdot 4^{2} - 64 \cdot 4 + 64$

Step 3.7.6.19.2.3.3

Multiply $- 3$ by $64$ .

$- 192 + 24 \cdot 4^{2} - 64 \cdot 4 + 64$

Step 3.7.6.19.2.3.4

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

$- 192 + 24 \cdot 16 - 64 \cdot 4 + 64$

Step 3.7.6.19.2.3.5

Multiply $24$ by $16$ .

$- 192 + 384 - 64 \cdot 4 + 64$

Step 3.7.6.19.2.3.6

Add $- 192$ and $384$ .

$192 - 64 \cdot 4 + 64$

Step 3.7.6.19.2.3.7

Multiply $- 64$ by $4$ .

$192 - 256 + 64$

Step 3.7.6.19.2.3.8

Subtract $256$ from $192$ .

$- 64 + 64$

Step 3.7.6.19.2.3.9

Add $- 64$ and $64$ .

$0$

Step 3.7.6.19.2.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 3 x^{3} + 24 x^{2} - 64 x + 64}{x - 4}$

Step 3.7.6.19.2.5

Divide $- 3 x^{3} + 24 x^{2} - 64 x + 64$ by $x - 4$ .

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Step 3.7.6.19.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$4$

$3 x^{3}$

$24 x^{2}$

$64 x$

$64$

Step 3.7.6.19.2.5.2

Divide the highest order term in the dividend $- 3 x^{3}$ by the highest order term in divisor $x$ .

				-	$3 x^{2}$
	$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$

Step 3.7.6.19.2.5.3

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			-	$3 x^{3}$	+	$12 x^{2}$

Step 3.7.6.19.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{3} + 12 x^{2}$

			-	$3 x^{2}$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$

Step 3.7.6.19.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$3 x^{2}$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$

Step 3.7.6.19.2.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$3 x^{2}$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$

Step 3.7.6.19.2.5.7

Divide the highest order term in the dividend $12 x^{2}$ by the highest order term in divisor $x$ .

			-	$3 x^{2}$	+	$12 x$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$

Step 3.7.6.19.2.5.8

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	+	$12 x$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					+	$12 x^{2}$	-	$48 x$

Step 3.7.6.19.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $12 x^{2} - 48 x$

			-	$3 x^{2}$	+	$12 x$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					-	$12 x^{2}$	+	$48 x$

Step 3.7.6.19.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$3 x^{2}$	+	$12 x$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					-	$12 x^{2}$	+	$48 x$
							-	$16 x$

Step 3.7.6.19.2.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$3 x^{2}$	+	$12 x$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					-	$12 x^{2}$	+	$48 x$
							-	$16 x$	+	$64$

Step 3.7.6.19.2.5.12

Divide the highest order term in the dividend $- 16 x$ by the highest order term in divisor $x$ .

			-	$3 x^{2}$	+	$12 x$	-	$16$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					-	$12 x^{2}$	+	$48 x$
							-	$16 x$	+	$64$

Step 3.7.6.19.2.5.13

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	+	$12 x$	-	$16$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					-	$12 x^{2}$	+	$48 x$
							-	$16 x$	+	$64$
							-	$16 x$	+	$64$

Step 3.7.6.19.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x + 64$

			-	$3 x^{2}$	+	$12 x$	-	$16$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					-	$12 x^{2}$	+	$48 x$
							-	$16 x$	+	$64$
							+	$16 x$	-	$64$

Step 3.7.6.19.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$3 x^{2}$	+	$12 x$	-	$16$
$x$	-	$4$	-	$3 x^{3}$	+	$24 x^{2}$	-	$64 x$	+	$64$
			+	$3 x^{3}$	-	$12 x^{2}$
					+	$12 x^{2}$	-	$64 x$
					-	$12 x^{2}$	+	$48 x$
							-	$16 x$	+	$64$
							+	$16 x$	-	$64$
										$0$

Step 3.7.6.19.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$- 3 x^{2} + 12 x - 16$

Step 3.7.6.19.2.6

Write $- 3 x^{3} + 24 x^{2} - 64 x + 64$ as a set of factors.

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

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

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

Step 3.7.7

Combine terms.

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Step 3.7.7.1

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

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Step 3.7.7.1.1

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

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

Step 3.7.7.1.2

Multiply $2$ by $2$ .

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

Step 3.7.7.2

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

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Step 3.7.7.2.1

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

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

Step 3.7.7.2.2

Multiply $2$ by $2$ .

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

Step 3.7.7.3

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

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Step 3.7.7.3.1

Factor $x$ out of $16 x (x - 4) (- 3 x^{2} + 12 x - 16)$ .

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

Step 3.7.7.3.2

Cancel the common factors.

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Step 3.7.7.3.2.1

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

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

Step 3.7.7.3.2.2

Cancel the common factor.

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

Step 3.7.7.3.2.3

Rewrite the expression.

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

Step 3.7.7.4

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

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Step 3.7.7.4.1

Factor $x - 4$ out of $16 (x - 4) (- 3 x^{2} + 12 x - 16)$ .

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

Step 3.7.7.4.2

Cancel the common factors.

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Step 3.7.7.4.2.1

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

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

Step 3.7.7.4.2.2

Cancel the common factor.

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

Step 3.7.7.4.2.3

Rewrite the expression.

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

Step 3.7.8

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 3.7.9

Factor $- 1$ out of $12 x$ .

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

Step 3.7.10

Factor $- 1$ out of $- (3 x^{2}) - (- 12 x)$ .

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

Step 3.7.11

Rewrite $- 16$ as $- 1 (16)$ .

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

Step 3.7.12

Factor $- 1$ out of $- (3 x^{2} - 12 x) - 1 (16)$ .

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

Step 3.7.13

Rewrite $- (3 x^{2} - 12 x + 16)$ as $- 1 (3 x^{2} - 12 x + 16)$ .

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

Step 3.7.14

Move the negative in front of the fraction.

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

Step 3.7.15

Multiply $- 1$ by $- 1$ .

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

Step 3.7.16

Multiply $\frac{16 (3 x^{2} - 12 x + 16)}{x^{3} {(x - 4)}^{3}}$ by $1$ .

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

Step 4

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

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate using the Constant Multiple Rule.

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Step 5.1.1.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x^{2} - 4 x}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x^{2} - 4 x}]$ .

$8 \frac{d}{d x} [\frac{1}{x^{2} - 4 x}]$

Step 5.1.1.2

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

$8 \frac{d}{d x} [{(x^{2} - 4 x)}^{- 1}]$

Step 5.1.2

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

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Step 5.1.2.1

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

$8 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 4 x])$

Step 5.1.2.2

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

$8 (- u^{- 2} \frac{d}{d x} [x^{2} - 4 x])$

Step 5.1.2.3

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

$8 (- {(x^{2} - 4 x)}^{- 2} \frac{d}{d x} [x^{2} - 4 x])$

Step 5.1.3

Differentiate.

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Step 5.1.3.1

Multiply $- 1$ by $8$ .

$- 8 ({(x^{2} - 4 x)}^{- 2} \frac{d}{d x} [x^{2} - 4 x])$

Step 5.1.3.2

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

$- 8 {(x^{2} - 4 x)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x])$

Step 5.1.3.3

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

$- 8 {(x^{2} - 4 x)}^{- 2} (2 x + \frac{d}{d x} [- 4 x])$

Step 5.1.3.4

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

$- 8 {(x^{2} - 4 x)}^{- 2} (2 x - 4 \frac{d}{d x} [x])$

Step 5.1.3.5

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

$- 8 {(x^{2} - 4 x)}^{- 2} (2 x - 4 \cdot 1)$

Step 5.1.3.6

Multiply $- 4$ by $1$ .

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

Step 5.1.4

Simplify.

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Step 5.1.4.1

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

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

Step 5.1.4.2

Combine terms.

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Step 5.1.4.2.1

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

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

Step 5.1.4.2.2

Move the negative in front of the fraction.

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

Step 5.1.4.3

Reorder the factors of $- \frac{8}{{(x^{2} - 4 x)}^{2}} (2 x - 4)$ .

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

Step 5.1.4.4

Apply the distributive property.

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

Step 5.1.4.5

Multiply $2$ by $- 1$ .

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

Step 5.1.4.6

Multiply $- 1$ by $- 4$ .

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

Step 5.1.4.7

Simplify the denominator.

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Step 5.1.4.7.1

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

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Step 5.1.4.7.1.1

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

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

Step 5.1.4.7.1.2

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

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

Step 5.1.4.7.1.3

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

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

Step 5.1.4.7.2

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

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

Step 5.1.4.8

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

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

Step 5.1.4.9

Simplify the numerator.

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Step 5.1.4.9.1

Factor $2$ out of $- 2 x + 4$ .

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Step 5.1.4.9.1.1

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

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

Step 5.1.4.9.1.2

Factor $2$ out of $4$ .

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

Step 5.1.4.9.1.3

Factor $2$ out of $2 (- x) + 2 (2)$ .

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

Step 5.1.4.9.2

Multiply $8$ by $2$ .

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

Step 5.1.4.10

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

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

Step 5.1.4.11

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

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

Step 5.1.4.12

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 5.1.4.13

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

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

Step 5.1.4.14

Move the negative in front of the fraction.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$16 (x - 2) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $16 (x - 2) = 0$ by $16$ and simplify.

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Step 6.3.1.1

Divide each term in $16 (x - 2) = 0$ by $16$ .

$\frac{16 (x - 2)}{16} = \frac{0}{16}$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $16$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

$\frac{16 (x - 2)}{16} = \frac{0}{16}$

Step 6.3.1.2.1.2

Divide $x - 2$ by $1$ .

$x - 2 = \frac{0}{16}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $16$ .

$x - 2 = 0$

Step 6.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

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

Step 7.2

Solve for $x$ .

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Step 7.2.1

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

$x^{2} = 0$

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

Step 7.2.2

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

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Step 7.2.2.1

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

$x^{2} = 0$

Step 7.2.2.2

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

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Step 7.2.2.2.1

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

$x = \pm \sqrt{0}$

Step 7.2.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.2.2.2.2.1

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

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

Step 7.2.2.2.2.2

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

$x = \pm 0$

Step 7.2.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7.2.3

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

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Step 7.2.3.1

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

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

Step 7.2.3.2

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

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Step 7.2.3.2.1

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

$x - 4 = 0$

Step 7.2.3.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7.2.4

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

$x = 0, 4$

Step 7.3

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

$x = 0, x = 4$

Step 8

Critical points to evaluate.

$x = 2$

Step 9

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

$\frac{16 (3 {(2)}^{2} - 12 \cdot 2 + 16)}{{(2)}^{3} {((2) - 4)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$\frac{16 (3 \cdot 4 - 12 \cdot 2 + 16)}{2^{3} {(2 - 4)}^{3}}$

Step 10.1.2

Multiply $3$ by $4$ .

$\frac{16 (12 - 12 \cdot 2 + 16)}{2^{3} {(2 - 4)}^{3}}$

Step 10.1.3

Multiply $- 12$ by $2$ .

$\frac{16 (12 - 24 + 16)}{2^{3} {(2 - 4)}^{3}}$

Step 10.1.4

Subtract $24$ from $12$ .

$\frac{16 (- 12 + 16)}{2^{3} {(2 - 4)}^{3}}$

Step 10.1.5

Add $- 12$ and $16$ .

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

Step 10.2

Simplify the denominator.

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Step 10.2.1

Subtract $4$ from $2$ .

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

Step 10.2.2

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

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

Step 10.2.3

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

$\frac{16 \cdot 4}{8 \cdot - 8}$

Step 10.3

Simplify the expression.

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Step 10.3.1

Multiply $16$ by $4$ .

$\frac{64}{8 \cdot - 8}$

Step 10.3.2

Multiply $8$ by $- 8$ .

$\frac{64}{- 64}$

Step 10.3.3

Divide $64$ by $- 64$ .

$- 1$

Step 11

$x = 2$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 2$ is a local maximum

Step 12

Find the y-value when $x = 2$ .

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Step 12.1

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

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

Step 12.2

Simplify the result.

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Step 12.2.1

Reduce the expression by cancelling the common factors.

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Step 12.2.1.1

Cancel the common factor of $8$ and ${(2)}^{2} - 4 \cdot 2$ .

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Step 12.2.1.1.1

Factor $2$ out of $8$ .

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

Step 12.2.1.1.2

Cancel the common factors.

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Step 12.2.1.1.2.1

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

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

Step 12.2.1.1.2.2

Factor $2$ out of $- 4 \cdot 2$ .

$f (2) = \frac{2 \cdot 4}{2 \cdot 2 + 2 (- 2 \cdot 2)}$

Step 12.2.1.1.2.3

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2)$ .

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

Step 12.2.1.1.2.4

Cancel the common factor.

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

Step 12.2.1.1.2.5

Rewrite the expression.

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

Step 12.2.1.2

Cancel the common factor of $4$ and $2 - 2 \cdot 2$ .

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Step 12.2.1.2.1

Factor $2$ out of $4$ .

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

Step 12.2.1.2.2

Cancel the common factors.

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Step 12.2.1.2.2.1

Factor $2$ out of $2$ .

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

Step 12.2.1.2.2.2

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

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

Step 12.2.1.2.2.3

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2)$ .

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

Step 12.2.1.2.2.4

Cancel the common factor.

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

Step 12.2.1.2.2.5

Rewrite the expression.

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

Step 12.2.2

Simplify the denominator.

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Step 12.2.2.1

Multiply $- 1$ by $2$ .

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

Step 12.2.2.2

Subtract $2$ from $1$ .

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

Step 12.2.3

Divide $2$ by $- 1$ .

$f (2) = - 2$

Step 12.2.4

The final answer is $- 2$ .

$y = - 2$

Step 13

These are the local extrema for $f (x) = \frac{8}{x^{2} - 4 x}$ .

$(2, - 2)$ is a local maxima

Step 14