Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

$x^{- 4} e^{x} + e^{x} (- 4 x^{- 5})$

Step 1.4

Simplify.

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Step 1.4.1

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

$\frac{1}{x^{4}} e^{x} + e^{x} (- 4 x^{- 5})$

Step 1.4.2

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

$\frac{1}{x^{4}} e^{x} + e^{x} (- 4 \frac{1}{x^{5}})$

Step 1.4.3

Combine terms.

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Step 1.4.3.1

Combine $\frac{1}{x^{4}}$ and $e^{x}$ .

$\frac{e^{x}}{x^{4}} + e^{x} (- 4 \frac{1}{x^{5}})$

Step 1.4.3.2

Combine $- 4$ and $\frac{1}{x^{5}}$ .

$\frac{e^{x}}{x^{4}} + e^{x} \frac{- 4}{x^{5}}$

Step 1.4.3.3

Move the negative in front of the fraction.

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

Step 1.4.3.4

Combine $e^{x}$ and $\frac{4}{x^{5}}$ .

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

Step 1.4.3.5

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

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

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{e^{x}}{x^{4}} - \frac{4 e^{x}}{x^{5}}$ with respect to $x$ is $\frac{d}{d x} [\frac{e^{x}}{x^{4}}] + \frac{d}{d x} [- \frac{4 e^{x}}{x^{5}}]$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $4$ by $- 1$ .

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

Step 2.2.5

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

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Step 2.2.5.1

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

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

Step 2.2.5.2

Multiply $4$ by $2$ .

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

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{4 e^{x}}{x^{5}}]$ .

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Step 2.3.1

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

$f'' (x) = \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}} - 4 \frac{d}{d x} \frac{e^{x}}{x^{5}}$

Step 2.3.2

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $5$ by $- 1$ .

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

Step 2.3.6

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

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Step 2.3.6.1

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

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

Step 2.3.6.2

Multiply $5$ by $2$ .

$f'' (x) = \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}} - 4 \frac{x^{5} e^{x} - 5 e^{x} x^{4}}{x^{10}}$

Step 2.3.7

Combine $- 4$ and $\frac{x^{5} e^{x} - 5 e^{x} x^{4}}{x^{10}}$ .

$f'' (x) = \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}} + \frac{- 4 (x^{5} e^{x} - 5 e^{x} x^{4})}{x^{10}}$

Step 2.3.8

Move the negative in front of the fraction.

$f'' (x) = \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}} - \frac{4 (x^{5} e^{x} - 5 e^{x} x^{4})}{x^{10}}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}} - \frac{4 (x^{5} e^{x}) + 4 (- 5 e^{x} x^{4})}{x^{10}}$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $- 5$ by $4$ .

$f'' (x) = \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}} - \frac{4 x^{5} e^{x} - 20 (e^{x} x^{4})}{x^{10}}$

Step 2.4.2.2

To write $\frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$f'' (x) = \frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}} \cdot \frac{x^{2}}{x^{2}} - \frac{4 x^{5} e^{x} - 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.2.3

Write each expression with a common denominator of $x^{10}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.2.3.1

Multiply $\frac{x^{4} e^{x} - 4 e^{x} x^{3}}{x^{8}}$ by $\frac{x^{2}}{x^{2}}$ .

$f'' (x) = \frac{(x^{4} e^{x} - 4 e^{x} x^{3}) x^{2}}{x^{8} x^{2}} - \frac{4 x^{5} e^{x} - 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.2.3.2

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

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Step 2.4.2.3.2.1

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

$f'' (x) = \frac{(x^{4} e^{x} - 4 e^{x} x^{3}) x^{2}}{x^{8 + 2}} - \frac{4 x^{5} e^{x} - 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.2.3.2.2

Add $8$ and $2$ .

$f'' (x) = \frac{(x^{4} e^{x} - 4 e^{x} x^{3}) x^{2}}{x^{10}} - \frac{4 x^{5} e^{x} - 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.2.4

Combine the numerators over the common denominator.

$f'' (x) = \frac{(x^{4} e^{x} - 4 e^{x} x^{3}) x^{2} - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.3

Reorder terms.

$f'' (x) = \frac{x^{2} (x^{4} e^{x} - 4 e^{x} x^{3}) - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4

Simplify the numerator.

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Step 2.4.4.1

Apply the distributive property.

$f'' (x) = \frac{x^{2} (x^{4} e^{x}) + x^{2} (- 4 e^{x} x^{3}) - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.2

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

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Step 2.4.4.2.1

Move $x^{4}$ .

$f'' (x) = \frac{x^{4} x^{2} e^{x} + x^{2} (- 4 e^{x} x^{3}) - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.2.2

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

$f'' (x) = \frac{x^{4 + 2} e^{x} + x^{2} (- 4 e^{x} x^{3}) - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.2.3

Add $4$ and $2$ .

$f'' (x) = \frac{x^{6} e^{x} + x^{2} (- 4 e^{x} x^{3}) - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{x^{6} e^{x} - 4 x^{2} (e^{x} x^{3}) - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.4

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

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Step 2.4.4.4.1

Move $x^{3}$ .

$f'' (x) = \frac{x^{6} e^{x} - 4 (x^{3} x^{2}) e^{x} - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.4.2

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

$f'' (x) = \frac{x^{6} e^{x} - 4 x^{3 + 2} e^{x} - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.4.3

Add $3$ and $2$ .

$f'' (x) = \frac{x^{6} e^{x} - 4 x^{5} e^{x} - (4 x^{5} e^{x} - 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.5

Apply the distributive property.

$f'' (x) = \frac{x^{6} e^{x} - 4 x^{5} e^{x} - (4 x^{5} e^{x}) - (- 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.6

Multiply $4$ by $- 1$ .

$f'' (x) = \frac{x^{6} e^{x} - 4 x^{5} e^{x} - 4 (x^{5} e^{x}) - (- 20 e^{x} x^{4})}{x^{10}}$

Step 2.4.4.7

Multiply $- 20$ by $- 1$ .

$f'' (x) = \frac{x^{6} e^{x} - 4 x^{5} e^{x} - 4 (x^{5} e^{x}) + 20 (e^{x} x^{4})}{x^{10}}$

Step 2.4.4.8

Remove parentheses.

$f'' (x) = \frac{x^{6} e^{x} - 4 x^{5} e^{x} - 4 x^{5} e^{x} + 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.4.9

Subtract $4 x^{5} e^{x}$ from $- 4 x^{5} e^{x}$ .

$f'' (x) = \frac{x^{6} e^{x} - 8 x^{5} e^{x} + 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.4.10

Factor $x^{4} e^{x}$ out of $x^{6} e^{x} - 8 x^{5} e^{x} + 20 e^{x} x^{4}$ .

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Step 2.4.4.10.1

Factor $x^{4} e^{x}$ out of $x^{6} e^{x}$ .

$f'' (x) = \frac{x^{4} e^{x} (x^{2}) - 8 x^{5} e^{x} + 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.4.10.2

Factor $x^{4} e^{x}$ out of $- 8 x^{5} e^{x}$ .

$f'' (x) = \frac{x^{4} e^{x} (x^{2}) + x^{4} e^{x} (- 8 x) + 20 e^{x} x^{4}}{x^{10}}$

Step 2.4.4.10.3

Factor $x^{4} e^{x}$ out of $20 e^{x} x^{4}$ .

$f'' (x) = \frac{x^{4} e^{x} (x^{2}) + x^{4} e^{x} (- 8 x) + x^{4} e^{x} (20)}{x^{10}}$

Step 2.4.4.10.4

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

$f'' (x) = \frac{x^{4} e^{x} (x^{2} - 8 x) + x^{4} e^{x} (20)}{x^{10}}$

Step 2.4.4.10.5

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

$f'' (x) = \frac{x^{4} e^{x} (x^{2} - 8 x + 20)}{x^{10}}$

Step 2.4.5

Cancel the common factor of $x^{4}$ and $x^{10}$ .

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Step 2.4.5.1

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

$f'' (x) = \frac{x^{4} (e^{x} (x^{2} - 8 x + 20))}{x^{10}}$

Step 2.4.5.2

Cancel the common factors.

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Step 2.4.5.2.1

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

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

Step 2.4.5.2.2

Cancel the common factor.

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

Step 2.4.5.2.3

Rewrite the expression.

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

Step 3

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

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

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

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

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

Step 4.1.2

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

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

Step 4.1.3

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

$x^{- 4} e^{x} + e^{x} (- 4 x^{- 5})$

Step 4.1.4

Simplify.

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Step 4.1.4.1

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

$\frac{1}{x^{4}} e^{x} + e^{x} (- 4 x^{- 5})$

Step 4.1.4.2

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

$\frac{1}{x^{4}} e^{x} + e^{x} (- 4 \frac{1}{x^{5}})$

Step 4.1.4.3

Combine terms.

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Step 4.1.4.3.1

Combine $\frac{1}{x^{4}}$ and $e^{x}$ .

$\frac{e^{x}}{x^{4}} + e^{x} (- 4 \frac{1}{x^{5}})$

Step 4.1.4.3.2

Combine $- 4$ and $\frac{1}{x^{5}}$ .

$\frac{e^{x}}{x^{4}} + e^{x} \frac{- 4}{x^{5}}$

Step 4.1.4.3.3

Move the negative in front of the fraction.

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

Step 4.1.4.3.4

Combine $e^{x}$ and $\frac{4}{x^{5}}$ .

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

Step 4.1.4.3.5

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

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

Step 4.2

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

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

Step 5

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

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Step 5.1

Set the first derivative equal to $0$ .

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

Step 5.2

Move $- \frac{4 e^{x}}{x^{5}}$ to the right side of the equation by adding it to both sides.

$\frac{e^{x}}{x^{4}} = \frac{4 e^{x}}{x^{5}}$

Step 5.3

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

$\frac{e^{x}}{x^{4}} x^{4} = \frac{4 e^{x}}{x^{5}} x^{4}$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

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

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Step 5.4.1.1.1

Cancel the common factor.

$\frac{e^{x}}{x^{4}} x^{4} = \frac{4 e^{x}}{x^{5}} x^{4}$

Step 5.4.1.1.2

Rewrite the expression.

$e^{x} = \frac{4 e^{x}}{x^{5}} x^{4}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

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

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Step 5.4.2.1.1

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

$e^{x} = \frac{4 e^{x}}{x^{4} x} x^{4}$

Step 5.4.2.1.2

Cancel the common factor.

$e^{x} = \frac{4 e^{x}}{x^{4} x} x^{4}$

Step 5.4.2.1.3

Rewrite the expression.

$e^{x} = \frac{4 e^{x}}{x}$

Step 5.5

Solve for $x$ .

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Step 5.5.1

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

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

Step 5.5.2

Expand the left side.

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Step 5.5.2.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

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

Step 5.5.2.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (\frac{4 e^{x}}{x})$

Step 5.5.2.3

Multiply $x$ by $1$ .

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

Step 5.5.3

Expand the right side.

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Step 5.5.3.1

Rewrite $\ln (\frac{4 e^{x}}{x})$ as $\ln (4 e^{x}) - \ln (x)$ .

$x = \ln (4 e^{x}) - \ln (x)$

Step 5.5.3.2

Rewrite $\ln (4 e^{x})$ as $\ln (4) + \ln (e^{x})$ .

$x = \ln (4) + \ln (e^{x}) - \ln (x)$

Step 5.5.3.3

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x = \ln (4) + x \ln (e) - \ln (x)$

Step 5.5.3.4

The natural logarithm of $e$ is $1$ .

$x = \ln (4) + x \cdot 1 - \ln (x)$

Step 5.5.3.5

Multiply $x$ by $1$ .

$x = \ln (4) + x - \ln (x)$

Step 5.5.4

Simplify the right side.

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Step 5.5.4.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x = x + \ln (\frac{4}{x})$

Step 5.5.5

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x + \ln (\frac{4}{x}) = x$

Step 5.5.6

Move all terms containing $x$ to the left side of the equation.

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Step 5.5.6.1

Subtract $x$ from both sides of the equation.

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

Step 5.5.6.2

Combine the opposite terms in $x + \ln (\frac{4}{x}) - x$ .

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Step 5.5.6.2.1

Subtract $x$ from $x$ .

$0 + \ln (\frac{4}{x}) = 0$

Step 5.5.6.2.2

Add $0$ and $\ln (\frac{4}{x})$ .

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

Step 5.5.7

To solve for $x$ , rewrite the equation using properties of logarithms.

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

Step 5.5.8

Rewrite $\ln (\frac{4}{x}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = \frac{4}{x}$

Step 5.5.9

Solve for $x$ .

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Step 5.5.9.1

Rewrite the equation as $\frac{4}{x} = e^{0}$ .

$\frac{4}{x} = e^{0}$

Step 5.5.9.2

Anything raised to $0$ is $1$ .

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

Step 5.5.9.3

Find the LCD of the terms in the equation.

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Step 5.5.9.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 5.5.9.3.2

The LCM of one and any expression is the expression.

$x$

Step 5.5.9.4

Multiply each term in $\frac{4}{x} = 1$ by $x$ to eliminate the fractions.

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Step 5.5.9.4.1

Multiply each term in $\frac{4}{x} = 1$ by $x$ .

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

Step 5.5.9.4.2

Simplify the left side.

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Step 5.5.9.4.2.1

Cancel the common factor of $x$ .

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Step 5.5.9.4.2.1.1

Cancel the common factor.

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

Step 5.5.9.4.2.1.2

Rewrite the expression.

$4 = 1 x$

Step 5.5.9.4.3

Simplify the right side.

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Step 5.5.9.4.3.1

Multiply $x$ by $1$ .

$4 = x$

Step 5.5.9.5

Rewrite the equation as $x = 4$ .

$x = 4$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

$x^{4} = 0$

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.

$x = \pm \sqrt[4]{0}$

Step 6.2.2

Simplify $\pm \sqrt[4]{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 6.2.2.2

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

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.3

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

$x^{5} = 0$

Step 6.4

Solve for $x$ .

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Step 6.4.1

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

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

Step 6.4.2

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

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Step 6.4.2.1

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

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

Step 6.4.2.2

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.

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

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

$\frac{e^{4} (16 - 8 \cdot 4 + 20)}{4^{6}}$

Step 9.1.2

Multiply $- 8$ by $4$ .

$\frac{e^{4} (16 - 32 + 20)}{4^{6}}$

Step 9.1.3

Subtract $32$ from $16$ .

$\frac{e^{4} (- 16 + 20)}{4^{6}}$

Step 9.1.4

Add $- 16$ and $20$ .

$\frac{e^{4} \cdot 4}{4^{6}}$

Step 9.2

Reduce the expression by cancelling the common factors.

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Step 9.2.1

Raise $4$ to the power of $6$ .

$\frac{e^{4} \cdot 4}{4096}$

Step 9.2.2

Cancel the common factor of $4$ and $4096$ .

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Step 9.2.2.1

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

$\frac{4 \cdot e^{4}}{4096}$

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $4$ out of $4096$ .

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

Step 9.2.2.2.2

Cancel the common factor.

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

Step 9.2.2.2.3

Rewrite the expression.

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

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.

$f (4) = {(4)}^{- 4} e^{4}$

Step 11.2

Simplify the result.

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Step 11.2.1

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

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

Step 11.2.2

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

$f (4) = \frac{1}{256} \cdot e^{4}$

Step 11.2.3

Combine $\frac{1}{256}$ and $e^{4}$ .

$f (4) = \frac{e^{4}}{256}$

Step 11.2.4

The final answer is $\frac{e^{4}}{256}$ .

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

Step 12

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

$(4, \frac{e^{4}}{256})$ is a local minima

Step 13