Calculus Examples

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

Step 1

Write $\frac{4 x}{{(x + 4)}^{2}}$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

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

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

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 $f (x) = x$ and $g (x) = {(x + 4)}^{2}$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $2$ by $2$ .

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

Step 2.3.2

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

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

Step 2.3.3

Multiply ${(x + 4)}^{2}$ by $1$ .

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

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 $g (x) = x + 4$ .

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $2$ by $- 1$ .

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

Step 2.5.2

Factor $x + 4$ out of ${(x + 4)}^{2} - 2 x ((x + 4) \frac{d}{d x} [x + 4])$ .

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Step 2.5.2.1

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

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

Step 2.5.2.2

Factor $x + 4$ out of $- 2 x ((x + 4) \frac{d}{d x} [x + 4])$ .

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

Step 2.5.2.3

Factor $x + 4$ out of $(x + 4) (x + 4) + (x + 4) (- 2 x (\frac{d}{d x} [x + 4]))$ .

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

Step 2.6

Cancel the common factors.

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Step 2.6.1

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

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

Step 2.6.2

Cancel the common factor.

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

Step 2.6.3

Rewrite the expression.

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

Step 2.7

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

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

Step 2.8

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

$4 \frac{x + 4 - 2 x (1 + \frac{d}{d x} [4])}{{(x + 4)}^{3}}$

Step 2.9

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

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

Step 2.10

Simplify terms.

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Step 2.10.1

Add $1$ and $0$ .

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

Step 2.10.2

Multiply $- 2$ by $1$ .

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

Step 2.10.3

Subtract $2 x$ from $x$ .

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

Step 2.10.4

Combine $4$ and $\frac{- x + 4}{{(x + 4)}^{3}}$ .

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

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

$\frac{4 (- x) + 4 \cdot 4}{{(x + 4)}^{3}}$

Step 2.11.2

Simplify each term.

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Step 2.11.2.1

Multiply $- 1$ by $4$ .

$\frac{- 4 x + 4 \cdot 4}{{(x + 4)}^{3}}$

Step 2.11.2.2

Multiply $4$ by $4$ .

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

Step 2.11.3

Factor $4$ out of $- 4 x + 16$ .

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Step 2.11.3.1

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

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

Step 2.11.3.2

Factor $4$ out of $16$ .

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

Step 2.11.3.3

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

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

Step 2.11.4

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

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

Step 2.11.5

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

$\frac{4 (- (x) - 1 (- 4))}{{(x + 4)}^{3}}$

Step 2.11.6

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

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

Step 2.11.7

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

$\frac{4 (- 1 (x - 4))}{{(x + 4)}^{3}}$

Step 2.11.8

Move the negative in front of the fraction.

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

Step 3

Find the second derivative of the function.

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Step 3.1

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

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

Step 3.2

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

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(x + 4)}^{3})}^{2}$ .

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Step 3.3.1.1

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

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

Step 3.3.1.2

Multiply $3$ by $2$ .

$f'' (x) = - 4 \frac{{(x + 4)}^{3} \frac{d}{d x} (x - 4) - (x - 4) \frac{d}{d x} {(x + 4)}^{3}}{{(x + 4)}^{6}}$

Step 3.3.2

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

$f'' (x) = - 4 \frac{{(x + 4)}^{3} (\frac{d}{d x} (x) + \frac{d}{d x} (- 4)) - (x - 4) \frac{d}{d x} {(x + 4)}^{3}}{{(x + 4)}^{6}}$

Step 3.3.3

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

$f'' (x) = - 4 \frac{{(x + 4)}^{3} (1 + \frac{d}{d x} (- 4)) - (x - 4) \frac{d}{d x} {(x + 4)}^{3}}{{(x + 4)}^{6}}$

Step 3.3.4

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

$f'' (x) = - 4 \frac{{(x + 4)}^{3} (1 + 0) - (x - 4) \frac{d}{d x} {(x + 4)}^{3}}{{(x + 4)}^{6}}$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$f'' (x) = - 4 \frac{{(x + 4)}^{3} \cdot 1 - (x - 4) \frac{d}{d x} {(x + 4)}^{3}}{{(x + 4)}^{6}}$

Step 3.3.5.2

Multiply ${(x + 4)}^{3}$ by $1$ .

$f'' (x) = - 4 \frac{{(x + 4)}^{3} - (x - 4) \frac{d}{d x} {(x + 4)}^{3}}{{(x + 4)}^{6}}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x + 4$ .

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Step 3.4.1

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

$f'' (x) = - 4 \frac{{(x + 4)}^{3} - (x - 4) (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x + 4))}{{(x + 4)}^{6}}$

Step 3.4.2

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

$f'' (x) = - 4 \frac{{(x + 4)}^{3} - (x - 4) (3 u^{2} \frac{d}{d x} (x + 4))}{{(x + 4)}^{6}}$

Step 3.4.3

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

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

Step 3.5

Simplify with factoring out.

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Step 3.5.1

Multiply $3$ by $- 1$ .

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

Step 3.5.2

Factor ${(x + 4)}^{2}$ out of ${(x + 4)}^{3} - 3 (x - 4) ({(x + 4)}^{2} \frac{d}{d x} [x + 4])$ .

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Step 3.5.2.1

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

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

Step 3.5.2.2

Factor ${(x + 4)}^{2}$ out of $- 3 (x - 4) ({(x + 4)}^{2} \frac{d}{d x} [x + 4])$ .

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

Step 3.5.2.3

Factor ${(x + 4)}^{2}$ out of ${(x + 4)}^{2} (x + 4) + {(x + 4)}^{2} (- 3 (x - 4) (\frac{d}{d x} [x + 4]))$ .

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

Step 3.6

Cancel the common factors.

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Step 3.6.1

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

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

Step 3.6.2

Cancel the common factor.

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

Step 3.6.3

Rewrite the expression.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Combine fractions.

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Step 3.10.1

Add $1$ and $0$ .

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

Step 3.10.2

Multiply $- 3$ by $1$ .

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

Step 3.10.3

Combine $- 4$ and $\frac{x + 4 - 3 (x - 4)}{{(x + 4)}^{4}}$ .

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

Step 3.10.4

Move the negative in front of the fraction.

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

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

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

Step 3.11.2

Apply the distributive property.

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

Step 3.11.3

Simplify the numerator.

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Step 3.11.3.1

Simplify each term.

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Step 3.11.3.1.1

Multiply $4$ by $4$ .

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

Step 3.11.3.1.2

Multiply $- 3$ by $4$ .

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

Step 3.11.3.1.3

Multiply $4 (- 3 \cdot - 4)$ .

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Step 3.11.3.1.3.1

Multiply $- 3$ by $- 4$ .

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

Step 3.11.3.1.3.2

Multiply $4$ by $12$ .

$f'' (x) = - \frac{4 x + 16 - 12 x + 48}{{(x + 4)}^{4}}$

Step 3.11.3.2

Subtract $12 x$ from $4 x$ .

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

Step 3.11.3.3

Add $16$ and $48$ .

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

Step 3.11.4

Factor $8$ out of $- 8 x + 64$ .

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Step 3.11.4.1

Factor $8$ out of $- 8 x$ .

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

Step 3.11.4.2

Factor $8$ out of $64$ .

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

Step 3.11.4.3

Factor $8$ out of $8 (- x) + 8 (8)$ .

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

Step 3.11.5

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

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

Step 3.11.6

Rewrite $8$ as $- 1 (- 8)$ .

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

Step 3.11.7

Factor $- 1$ out of $- (x) - 1 (- 8)$ .

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

Step 3.11.8

Rewrite $- (x - 8)$ as $- 1 (x - 8)$ .

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

Step 3.11.9

Move the negative in front of the fraction.

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

Step 3.11.10

Multiply $- 1$ by $- 1$ .

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

Step 3.11.11

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

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

Step 4

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

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

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

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

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

Step 5.1.2

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

Step 5.1.3

Differentiate using the Power Rule.

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Step 5.1.3.1

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

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Step 5.1.3.1.1

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

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

Step 5.1.3.1.2

Multiply $2$ by $2$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Multiply ${(x + 4)}^{2}$ by $1$ .

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

Step 5.1.4

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 $g (x) = x + 4$ .

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Step 5.1.4.1

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

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

Step 5.1.4.2

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

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

Step 5.1.4.3

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

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

Step 5.1.5

Simplify with factoring out.

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Step 5.1.5.1

Multiply $2$ by $- 1$ .

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

Step 5.1.5.2

Factor $x + 4$ out of ${(x + 4)}^{2} - 2 x ((x + 4) \frac{d}{d x} [x + 4])$ .

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Step 5.1.5.2.1

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

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

Step 5.1.5.2.2

Factor $x + 4$ out of $- 2 x ((x + 4) \frac{d}{d x} [x + 4])$ .

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

Step 5.1.5.2.3

Factor $x + 4$ out of $(x + 4) (x + 4) + (x + 4) (- 2 x (\frac{d}{d x} [x + 4]))$ .

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

Step 5.1.6

Cancel the common factors.

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Step 5.1.6.1

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

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

Step 5.1.6.2

Cancel the common factor.

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

Step 5.1.6.3

Rewrite the expression.

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

Step 5.1.7

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

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

Step 5.1.8

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

$4 \frac{x + 4 - 2 x (1 + \frac{d}{d x} [4])}{{(x + 4)}^{3}}$

Step 5.1.9

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

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

Step 5.1.10

Simplify terms.

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Step 5.1.10.1

Add $1$ and $0$ .

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

Step 5.1.10.2

Multiply $- 2$ by $1$ .

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

Step 5.1.10.3

Subtract $2 x$ from $x$ .

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

Step 5.1.10.4

Combine $4$ and $\frac{- x + 4}{{(x + 4)}^{3}}$ .

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

Step 5.1.11

Simplify.

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Step 5.1.11.1

Apply the distributive property.

$\frac{4 (- x) + 4 \cdot 4}{{(x + 4)}^{3}}$

Step 5.1.11.2

Simplify each term.

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Step 5.1.11.2.1

Multiply $- 1$ by $4$ .

$\frac{- 4 x + 4 \cdot 4}{{(x + 4)}^{3}}$

Step 5.1.11.2.2

Multiply $4$ by $4$ .

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

Step 5.1.11.3

Factor $4$ out of $- 4 x + 16$ .

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Step 5.1.11.3.1

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

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

Step 5.1.11.3.2

Factor $4$ out of $16$ .

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

Step 5.1.11.3.3

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

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

Step 5.1.11.4

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

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

Step 5.1.11.5

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

$\frac{4 (- (x) - 1 (- 4))}{{(x + 4)}^{3}}$

Step 5.1.11.6

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

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

Step 5.1.11.7

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

$\frac{4 (- 1 (x - 4))}{{(x + 4)}^{3}}$

Step 5.1.11.8

Move the negative in front of the fraction.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$4 (x - 4) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $4 (x - 4) = 0$ by $4$ and simplify.

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Step 6.3.1.1

Divide each term in $4 (x - 4) = 0$ by $4$ .

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

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $4$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

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

Step 6.3.1.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{0}{4}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $4$ .

$x - 4 = 0$

Step 6.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

${(x + 4)}^{3} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

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

$x + 4 = 0$

Step 7.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 8

Critical points to evaluate.

$x = 4$

Step 9

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

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

Step 10

Evaluate the second derivative.

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Step 10.1

Subtract $8$ from $4$ .

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

Step 10.2

Simplify the denominator.

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Step 10.2.1

Add $4$ and $4$ .

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

Step 10.2.2

Raise $8$ to the power of $4$ .

$\frac{8 \cdot - 4}{4096}$

Step 10.3

Reduce the expression by cancelling the common factors.

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Step 10.3.1

Multiply $8$ by $- 4$ .

$\frac{- 32}{4096}$

Step 10.3.2

Cancel the common factor of $- 32$ and $4096$ .

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Step 10.3.2.1

Factor $32$ out of $- 32$ .

$\frac{32 (- 1)}{4096}$

Step 10.3.2.2

Cancel the common factors.

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Step 10.3.2.2.1

Factor $32$ out of $4096$ .

$\frac{32 \cdot - 1}{32 \cdot 128}$

Step 10.3.2.2.2

Cancel the common factor.

$\frac{32 \cdot - 1}{32 \cdot 128}$

Step 10.3.2.2.3

Rewrite the expression.

$\frac{- 1}{128}$

Step 10.3.3

Move the negative in front of the fraction.

$- \frac{1}{128}$

Step 11

$x = 4$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 4$ is a local maximum

Step 12

Find the y-value when $x = 4$ .

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Step 12.1

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

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

Step 12.2

Simplify the result.

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Step 12.2.1

Multiply $4$ by $4$ .

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

Step 12.2.2

Simplify the denominator.

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Step 12.2.2.1

Add $4$ and $4$ .

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

Step 12.2.2.2

Raise $8$ to the power of $2$ .

$f (4) = \frac{16}{64}$

Step 12.2.3

Cancel the common factor of $16$ and $64$ .

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Step 12.2.3.1

Factor $16$ out of $16$ .

$f (4) = \frac{16 (1)}{64}$

Step 12.2.3.2

Cancel the common factors.

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Step 12.2.3.2.1

Factor $16$ out of $64$ .

$f (4) = \frac{16 \cdot 1}{16 \cdot 4}$

Step 12.2.3.2.2

Cancel the common factor.

$f (4) = \frac{16 \cdot 1}{16 \cdot 4}$

Step 12.2.3.2.3

Rewrite the expression.

$f (4) = \frac{1}{4}$

Step 12.2.4

The final answer is $\frac{1}{4}$ .

$y = \frac{1}{4}$

Step 13

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

$(4, \frac{1}{4})$ is a local maxima

Step 14