Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

$4 \frac{d}{d x} [\frac{e^{x}}{x^{4}}]$

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

$4 \frac{x^{4} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{4}]}{x^{4 \cdot 2}}$

Step 1.3.2

Multiply $4$ by $2$ .

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

Step 1.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 1.5

Differentiate using the Power Rule.

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Step 1.5.1

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

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

Step 1.5.2

Combine fractions.

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Step 1.5.2.1

Multiply $4$ by $- 1$ .

$4 \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}}$

Step 1.5.2.2

Combine $4$ and $\frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}}$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Apply the distributive property.

$\frac{4 (x^{4} e^{x}) + 4 (- 4 e^{x} x^{3})}{x^{8}}$

Step 1.6.2

Simplify the numerator.

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Step 1.6.2.1

Multiply $- 4$ by $4$ .

$\frac{4 x^{4} e^{x} - 16 e^{x} x^{3}}{x^{8}}$

Step 1.6.2.2

Reorder factors in $4 x^{4} e^{x} - 16 e^{x} x^{3}$ .

$\frac{4 x^{4} e^{x} - 16 x^{3} e^{x}}{x^{8}}$

Step 1.6.3

Reorder terms.

$\frac{4 e^{x} x^{4} - 16 e^{x} x^{3}}{x^{8}}$

Step 1.6.4

Factor $4 e^{x} x^{3}$ out of $4 e^{x} x^{4} - 16 e^{x} x^{3}$ .

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Step 1.6.4.1

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

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

Step 1.6.4.2

Factor $4 e^{x} x^{3}$ out of $- 16 e^{x} x^{3}$ .

$\frac{4 e^{x} x^{3} (x) + 4 e^{x} x^{3} (- 4)}{x^{8}}$

Step 1.6.4.3

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

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

Step 1.6.5

Cancel the common factor of $x^{3}$ and $x^{8}$ .

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Step 1.6.5.1

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

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

Step 1.6.5.2

Cancel the common factors.

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Step 1.6.5.2.1

Factor $x^{3}$ out of $x^{8}$ .

$\frac{x^{3} (4 e^{x} (x - 4))}{x^{3} x^{5}}$

Step 1.6.5.2.2

Cancel the common factor.

$\frac{x^{3} (4 e^{x} (x - 4))}{x^{3} x^{5}}$

Step 1.6.5.2.3

Rewrite the expression.

$\frac{4 e^{x} (x - 4)}{x^{5}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 e^{x} (x - 4)}{x^{5}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{e^{x} (x - 4)}{x^{5}}]$ .

$f'' (x) = 4 \frac{d}{d x} (\frac{e^{x} (x - 4)}{x^{5}})$

Step 2.2

$f'' (x) = 4 (\frac{x^{5} \frac{d}{d x} (e^{x} (x - 4)) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{{(x^{5})}^{2}})$

Step 2.3

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 2.3.1

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

$f'' (x) = 4 (\frac{x^{5} \frac{d}{d x} (e^{x} (x - 4)) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{5 \cdot 2}})$

Step 2.3.2

Multiply $5$ by $2$ .

$f'' (x) = 4 (\frac{x^{5} \frac{d}{d x} (e^{x} (x - 4)) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.4

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 $f (x) = e^{x}$ and $g (x) = x - 4$ .

$f'' (x) = 4 (\frac{x^{5} (e^{x} \frac{d}{d x} (x - 4) + (x - 4) \frac{d}{d x} (e^{x})) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.5

Differentiate.

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Step 2.5.1

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

$f'' (x) = 4 (\frac{x^{5} (e^{x} (\frac{d}{d x} (x) + \frac{d}{d x} (- 4)) + (x - 4) \frac{d}{d x} (e^{x})) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.5.2

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

$f'' (x) = 4 (\frac{x^{5} (e^{x} (1 + \frac{d}{d x} (- 4)) + (x - 4) \frac{d}{d x} (e^{x})) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.5.3

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

$f'' (x) = 4 (\frac{x^{5} (e^{x} (1 + 0) + (x - 4) \frac{d}{d x} (e^{x})) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

$f'' (x) = 4 (\frac{x^{5} (e^{x} \cdot 1 + (x - 4) \frac{d}{d x} (e^{x})) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.5.4.2

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

$f'' (x) = 4 (\frac{x^{5} (e^{x} + (x - 4) \frac{d}{d x} (e^{x})) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.6

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$f'' (x) = 4 (\frac{x^{5} (e^{x} + (x - 4) e^{x}) - e^{x} (x - 4) \frac{d}{d x} x^{5}}{x^{10}})$

Step 2.7

Differentiate using the Power Rule.

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Step 2.7.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$f'' (x) = 4 (\frac{x^{5} (e^{x} + (x - 4) e^{x}) - e^{x} (x - 4) (5 x^{4})}{x^{10}})$

Step 2.7.2

Simplify with factoring out.

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Step 2.7.2.1

Multiply $5$ by $- 1$ .

$f'' (x) = 4 (\frac{x^{5} (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4) x^{4}}{x^{10}})$

Step 2.7.2.2

Factor $x^{4}$ out of $x^{5} (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4) x^{4}$ .

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Step 2.7.2.2.1

Factor $x^{4}$ out of $x^{5} (e^{x} + (x - 4) e^{x})$ .

$f'' (x) = 4 (\frac{x^{4} (x (e^{x} + (x - 4) e^{x})) - 5 e^{x} (x - 4) x^{4}}{x^{10}})$

Step 2.7.2.2.2

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

$f'' (x) = 4 (\frac{x^{4} (x (e^{x} + (x - 4) e^{x})) + x^{4} (- 5 e^{x} (x - 4))}{x^{10}})$

Step 2.7.2.2.3

Factor $x^{4}$ out of $x^{4} (x (e^{x} + (x - 4) e^{x})) + x^{4} (- 5 e^{x} (x - 4))$ .

$f'' (x) = 4 (\frac{x^{4} (x (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4))}{x^{10}})$

Step 2.8

Cancel the common factors.

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Step 2.8.1

Factor $x^{4}$ out of $x^{10}$ .

$f'' (x) = 4 (\frac{x^{4} (x (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4))}{x^{4} x^{6}})$

Step 2.8.2

Cancel the common factor.

$f'' (x) = 4 (\frac{x^{4} (x (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4))}{x^{4} x^{6}})$

Step 2.8.3

Rewrite the expression.

$f'' (x) = 4 (\frac{x (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4)}{x^{6}})$

Step 2.9

Combine $4$ and $\frac{x (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4)}{x^{6}}$ .

$f'' (x) = \frac{4 (x (e^{x} + (x - 4) e^{x}) - 5 e^{x} (x - 4))}{x^{6}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$f'' (x) = \frac{4 (x (e^{x} + x e^{x} - 4 e^{x}) - 5 e^{x} (x - 4))}{x^{6}}$

Step 2.10.2

Apply the distributive property.

$f'' (x) = \frac{4 (x e^{x} + x (x e^{x}) + x (- 4 e^{x}) - 5 e^{x} (x - 4))}{x^{6}}$

Step 2.10.3

Apply the distributive property.

$f'' (x) = \frac{4 (x e^{x} + x (x e^{x}) + x (- 4 e^{x}) - 5 e^{x} x - 5 e^{x} \cdot - 4)}{x^{6}}$

Step 2.10.4

Apply the distributive property.

$f'' (x) = \frac{4 (x e^{x}) + 4 (x (x e^{x})) + 4 (x (- 4 e^{x})) + 4 (- 5 e^{x} x) + 4 (- 5 e^{x} \cdot - 4)}{x^{6}}$

Step 2.10.5

Simplify the numerator.

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Step 2.10.5.1

Simplify each term.

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Step 2.10.5.1.1

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

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Step 2.10.5.1.1.1

Move $x$ .

$f'' (x) = \frac{4 x e^{x} + 4 (x \cdot x e^{x}) + 4 (x (- 4 e^{x})) + 4 (- 5 e^{x} x) + 4 (- 5 e^{x} \cdot - 4)}{x^{6}}$

Step 2.10.5.1.1.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x e^{x} + 4 (x^{2} e^{x}) + 4 (x (- 4 e^{x})) + 4 (- 5 e^{x} x) + 4 (- 5 e^{x} \cdot - 4)}{x^{6}}$

Step 2.10.5.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x e^{x} + 4 x^{2} e^{x} + 4 (- 4 x e^{x}) + 4 (- 5 e^{x} x) + 4 (- 5 e^{x} \cdot - 4)}{x^{6}}$

Step 2.10.5.1.3

Multiply $- 4$ by $4$ .

$f'' (x) = \frac{4 x e^{x} + 4 x^{2} e^{x} - 16 (x e^{x}) + 4 (- 5 e^{x} x) + 4 (- 5 e^{x} \cdot - 4)}{x^{6}}$

Step 2.10.5.1.4

Multiply $- 5$ by $4$ .

$f'' (x) = \frac{4 x e^{x} + 4 x^{2} e^{x} - 16 x e^{x} - 20 (e^{x} x) + 4 (- 5 e^{x} \cdot - 4)}{x^{6}}$

Step 2.10.5.1.5

Multiply $- 4$ by $- 5$ .

$f'' (x) = \frac{4 x e^{x} + 4 x^{2} e^{x} - 16 x e^{x} - 20 e^{x} x + 4 (20 e^{x})}{x^{6}}$

Step 2.10.5.1.6

Multiply $20$ by $4$ .

$f'' (x) = \frac{4 x e^{x} + 4 x^{2} e^{x} - 16 x e^{x} - 20 e^{x} x + 80 e^{x}}{x^{6}}$

Step 2.10.5.2

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

$f'' (x) = \frac{- 12 x e^{x} + 4 x^{2} e^{x} - 20 e^{x} x + 80 e^{x}}{x^{6}}$

Step 2.10.5.3

Subtract $20 e^{x} x$ from $- 12 x e^{x}$ .

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Step 2.10.5.3.1

Move $e^{x}$ .

$f'' (x) = \frac{- 12 x e^{x} - 20 x e^{x} + 4 x^{2} e^{x} + 80 e^{x}}{x^{6}}$

Step 2.10.5.3.2

Subtract $20 x e^{x}$ from $- 12 x e^{x}$ .

$f'' (x) = \frac{- 32 x e^{x} + 4 x^{2} e^{x} + 80 e^{x}}{x^{6}}$

Step 2.10.6

Reorder terms.

$f'' (x) = \frac{- 32 e^{x} x + 4 e^{x} x^{2} + 80 e^{x}}{x^{6}}$

Step 2.10.7

Simplify the numerator.

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Step 2.10.7.1

Factor $4 e^{x}$ out of $- 32 e^{x} x + 4 e^{x} x^{2} + 80 e^{x}$ .

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Step 2.10.7.1.1

Factor $4 e^{x}$ out of $- 32 e^{x} x$ .

$f'' (x) = \frac{4 e^{x} (- 8 x) + 4 e^{x} x^{2} + 80 e^{x}}{x^{6}}$

Step 2.10.7.1.2

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

$f'' (x) = \frac{4 e^{x} (- 8 x) + 4 e^{x} (x^{2}) + 80 e^{x}}{x^{6}}$

Step 2.10.7.1.3

Factor $4 e^{x}$ out of $80 e^{x}$ .

$f'' (x) = \frac{4 e^{x} (- 8 x) + 4 e^{x} (x^{2}) + 4 e^{x} (20)}{x^{6}}$

Step 2.10.7.1.4

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

$f'' (x) = \frac{4 e^{x} (- 8 x + x^{2}) + 4 e^{x} (20)}{x^{6}}$

Step 2.10.7.1.5

Factor $4 e^{x}$ out of $4 e^{x} (- 8 x + x^{2}) + 4 e^{x} (20)$ .

$f'' (x) = \frac{4 e^{x} (- 8 x + x^{2} + 20)}{x^{6}}$

Step 2.10.7.2

Reorder terms.

$f'' (x) = \frac{4 e^{x} (x^{2} - 8 x + 20)}{x^{6}}$

Step 3

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

$\frac{4 e^{x} (x - 4)}{x^{5}} = 0$

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

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

$f' (x) = 4 \frac{d}{d x} (\frac{e^{x}}{x^{4}})$

Step 4.1.2

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

Multiply $4$ by $2$ .

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

Step 4.1.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 4.1.5

Differentiate using the Power Rule.

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Step 4.1.5.1

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

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

Step 4.1.5.2

Combine fractions.

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Step 4.1.5.2.1

Multiply $4$ by $- 1$ .

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

Step 4.1.5.2.2

Combine $4$ and $\frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}}$ .

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

Step 4.1.6

Simplify.

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Step 4.1.6.1

Apply the distributive property.

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

Step 4.1.6.2

Simplify the numerator.

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Step 4.1.6.2.1

Multiply $- 4$ by $4$ .

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

Step 4.1.6.2.2

Reorder factors in $4 x^{4} e^{x} - 16 e^{x} x^{3}$ .

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

Step 4.1.6.3

Reorder terms.

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

Step 4.1.6.4

Factor $4 e^{x} x^{3}$ out of $4 e^{x} x^{4} - 16 e^{x} x^{3}$ .

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Step 4.1.6.4.1

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

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

Step 4.1.6.4.2

Factor $4 e^{x} x^{3}$ out of $- 16 e^{x} x^{3}$ .

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

Step 4.1.6.4.3

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

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

Step 4.1.6.5

Cancel the common factor of $x^{3}$ and $x^{8}$ .

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Step 4.1.6.5.1

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

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

Step 4.1.6.5.2

Cancel the common factors.

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Step 4.1.6.5.2.1

Factor $x^{3}$ out of $x^{8}$ .

$f' (x) = \frac{x^{3} (4 e^{x} (x - 4))}{x^{3} x^{5}}$

Step 4.1.6.5.2.2

Cancel the common factor.

$f' (x) = \frac{x^{3} (4 e^{x} (x - 4))}{x^{3} x^{5}}$

Step 4.1.6.5.2.3

Rewrite the expression.

$f' (x) = \frac{4 e^{x} (x - 4)}{x^{5}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4 e^{x} (x - 4)}{x^{5}}$ .

$\frac{4 e^{x} (x - 4)}{x^{5}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{4 e^{x} (x - 4)}{x^{5}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{4 e^{x} (x - 4)}{x^{5}} = 0$

Step 5.2

Set the numerator equal to zero.

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

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

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

$e^{x} = 0$

$x - 4 = 0$

Step 5.3.2

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

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Step 5.3.2.1

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

$e^{x} = 0$

Step 5.3.2.2

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

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Step 5.3.2.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (0)$

Step 5.3.2.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 5.3.2.2.3

There is no solution for $e^{x} = 0$

No solution

Step 5.3.3

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

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Step 5.3.3.1

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

$x - 4 = 0$

Step 5.3.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5.3.4

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

$x = 4$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{4 e^{x} (x - 4)}{x^{5}}$ equal to $0$ to find where the expression is undefined.

$x^{5} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

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

$x = \sqrt[5]{0}$

Step 6.2.2

Simplify $\sqrt[5]{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{5}$ .

$x = \sqrt[5]{0^{5}}$

Step 6.2.2.2

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

$x = 0$

Step 7

Critical points to evaluate.

$x = 4$

Step 8

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

$\frac{4 e^{4} ({(4)}^{2} - 8 \cdot 4 + 20)}{{(4)}^{6}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Cancel the common factor of $4$ and ${(4)}^{6}$ .

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Step 9.1.1

Factor $4$ out of $4 e^{4} ({(4)}^{2} - 8 \cdot 4 + 20)$ .

$\frac{4 (e^{4} ({(4)}^{2} - 8 \cdot 4 + 20))}{{(4)}^{6}}$

Step 9.1.2

Cancel the common factors.

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Step 9.1.2.1

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

$\frac{4 (e^{4} ({(4)}^{2} - 8 \cdot 4 + 20))}{4 \cdot 4^{5}}$

Step 9.1.2.2

Cancel the common factor.

$\frac{4 (e^{4} ({(4)}^{2} - 8 \cdot 4 + 20))}{4 \cdot 4^{5}}$

Step 9.1.2.3

Rewrite the expression.

$\frac{e^{4} ({(4)}^{2} - 8 \cdot 4 + 20)}{4^{5}}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

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

$\frac{e^{4} (16 - 8 \cdot 4 + 20)}{4^{5}}$

Step 9.2.2

Multiply $- 8$ by $4$ .

$\frac{e^{4} (16 - 32 + 20)}{4^{5}}$

Step 9.2.3

Subtract $32$ from $16$ .

$\frac{e^{4} (- 16 + 20)}{4^{5}}$

Step 9.2.4

Add $- 16$ and $20$ .

$\frac{e^{4} \cdot 4}{4^{5}}$

Step 9.3

Reduce the expression by cancelling the common factors.

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Step 9.3.1

Raise $4$ to the power of $5$ .

$\frac{e^{4} \cdot 4}{1024}$

Step 9.3.2

Cancel the common factor of $4$ and $1024$ .

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Step 9.3.2.1

Factor $4$ out of $e^{4} \cdot 4$ .

$\frac{4 \cdot e^{4}}{1024}$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $4$ out of $1024$ .

$\frac{4 \cdot e^{4}}{4 \cdot 256}$

Step 9.3.2.2.2

Cancel the common factor.

$\frac{4 \cdot e^{4}}{4 \cdot 256}$

Step 9.3.2.2.3

Rewrite the expression.

$\frac{e^{4}}{256}$

Step 10

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

$x = 4$ is a local minimum

Step 11

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

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Step 11.1

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

$f (4) = \frac{4 e^{4}}{{(4)}^{4}}$

Step 11.2

Simplify the result.

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Step 11.2.1

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

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Step 11.2.1.1

Factor $4$ out of $4 e^{4}$ .

$f (4) = \frac{4 (e^{4})}{{(4)}^{4}}$

Step 11.2.1.2

Cancel the common factors.

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Step 11.2.1.2.1

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

$f (4) = \frac{4 (e^{4})}{4 \cdot 4^{3}}$

Step 11.2.1.2.2

Cancel the common factor.

$f (4) = \frac{4 e^{4}}{4 \cdot 4^{3}}$

Step 11.2.1.2.3

Rewrite the expression.

$f (4) = \frac{e^{4}}{4^{3}}$

Step 11.2.2

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

$f (4) = \frac{e^{4}}{64}$

Step 11.2.3

The final answer is $\frac{e^{4}}{64}$ .

$y = \frac{e^{4}}{64}$

Step 12

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

$(4, \frac{e^{4}}{64})$ is a local minima

Step 13