Calculus Examples

$\frac{3 e^{x}}{x^{3}}$

Step 1

Write $\frac{3 e^{x}}{x^{3}}$ as a function.

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

Step 2

Find the first derivative of the function.

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3 e^{x}}{x^{3}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{e^{x}}{x^{3}}]$ .

$3 \frac{d}{d x} [\frac{e^{x}}{x^{3}}]$

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 $f (x) = e^{x}$ and $g (x) = x^{3}$ .

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

Step 2.3

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

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Step 2.3.1

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

$3 \frac{x^{3} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 2.3.2

Multiply $3$ by $2$ .

$3 \frac{x^{3} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.4

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

$3 \frac{x^{3} e^{x} - e^{x} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 2.5

Differentiate using the Power Rule.

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Step 2.5.1

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

$3 \frac{x^{3} e^{x} - e^{x} (3 x^{2})}{x^{6}}$

Step 2.5.2

Combine fractions.

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Step 2.5.2.1

Multiply $3$ by $- 1$ .

$3 \frac{x^{3} e^{x} - 3 e^{x} x^{2}}{x^{6}}$

Step 2.5.2.2

Combine $3$ and $\frac{x^{3} e^{x} - 3 e^{x} x^{2}}{x^{6}}$ .

$\frac{3 (x^{3} e^{x} - 3 e^{x} x^{2})}{x^{6}}$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$\frac{3 (x^{3} e^{x}) + 3 (- 3 e^{x} x^{2})}{x^{6}}$

Step 2.6.2

Simplify the numerator.

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Step 2.6.2.1

Multiply $- 3$ by $3$ .

$\frac{3 x^{3} e^{x} - 9 e^{x} x^{2}}{x^{6}}$

Step 2.6.2.2

Reorder factors in $3 x^{3} e^{x} - 9 e^{x} x^{2}$ .

$\frac{3 x^{3} e^{x} - 9 x^{2} e^{x}}{x^{6}}$

Step 2.6.3

Reorder terms.

$\frac{3 e^{x} x^{3} - 9 e^{x} x^{2}}{x^{6}}$

Step 2.6.4

Factor $3 e^{x} x^{2}$ out of $3 e^{x} x^{3} - 9 e^{x} x^{2}$ .

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Step 2.6.4.1

Factor $3 e^{x} x^{2}$ out of $3 e^{x} x^{3}$ .

$\frac{3 e^{x} x^{2} (x) - 9 e^{x} x^{2}}{x^{6}}$

Step 2.6.4.2

Factor $3 e^{x} x^{2}$ out of $- 9 e^{x} x^{2}$ .

$\frac{3 e^{x} x^{2} (x) + 3 e^{x} x^{2} (- 3)}{x^{6}}$

Step 2.6.4.3

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

$\frac{3 e^{x} x^{2} (x - 3)}{x^{6}}$

Step 2.6.5

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 2.6.5.1

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

$\frac{x^{2} (3 e^{x} (x - 3))}{x^{6}}$

Step 2.6.5.2

Cancel the common factors.

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Step 2.6.5.2.1

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

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

Step 2.6.5.2.2

Cancel the common factor.

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

Step 2.6.5.2.3

Rewrite the expression.

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

Step 3

Find the second derivative of the function.

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Step 3.1

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

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

Step 3.2

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $4$ by $2$ .

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

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 $f (x) = e^{x}$ and $g (x) = x - 3$ .

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

Step 3.5

Differentiate.

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Step 3.5.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

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

Step 3.5.2

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

$f'' (x) = 3 (\frac{x^{4} (e^{x} (1 + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} (e^{x})) - e^{x} (x - 3) \frac{d}{d x} x^{4}}{x^{8}})$

Step 3.5.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f'' (x) = 3 (\frac{x^{4} (e^{x} (1 + 0) + (x - 3) \frac{d}{d x} (e^{x})) - e^{x} (x - 3) \frac{d}{d x} x^{4}}{x^{8}})$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

Add $1$ and $0$ .

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

Step 3.5.4.2

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

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

Step 3.6

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

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

Step 3.7

Differentiate using the Power Rule.

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Step 3.7.1

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

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

Step 3.7.2

Simplify with factoring out.

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Step 3.7.2.1

Multiply $4$ by $- 1$ .

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

Step 3.7.2.2

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

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Step 3.7.2.2.1

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

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

Step 3.7.2.2.2

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

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

Step 3.7.2.2.3

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

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

Step 3.8

Cancel the common factors.

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Step 3.8.1

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

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

Step 3.8.2

Cancel the common factor.

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

Step 3.8.3

Rewrite the expression.

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

Step 3.9

Combine $3$ and $\frac{x (e^{x} + (x - 3) e^{x}) - 4 e^{x} (x - 3)}{x^{5}}$ .

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

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

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

Step 3.10.2

Apply the distributive property.

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

Step 3.10.3

Apply the distributive property.

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

Step 3.10.4

Apply the distributive property.

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

Step 3.10.5

Simplify the numerator.

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Step 3.10.5.1

Simplify each term.

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Step 3.10.5.1.1

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

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Step 3.10.5.1.1.1

Move $x$ .

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

Step 3.10.5.1.1.2

Multiply $x$ by $x$ .

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

Step 3.10.5.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.10.5.1.3

Multiply $- 3$ by $3$ .

$f'' (x) = \frac{3 x e^{x} + 3 x^{2} e^{x} - 9 (x e^{x}) + 3 (- 4 e^{x} x) + 3 (- 4 e^{x} \cdot - 3)}{x^{5}}$

Step 3.10.5.1.4

Multiply $- 4$ by $3$ .

$f'' (x) = \frac{3 x e^{x} + 3 x^{2} e^{x} - 9 x e^{x} - 12 (e^{x} x) + 3 (- 4 e^{x} \cdot - 3)}{x^{5}}$

Step 3.10.5.1.5

Multiply $- 3$ by $- 4$ .

$f'' (x) = \frac{3 x e^{x} + 3 x^{2} e^{x} - 9 x e^{x} - 12 e^{x} x + 3 (12 e^{x})}{x^{5}}$

Step 3.10.5.1.6

Multiply $12$ by $3$ .

$f'' (x) = \frac{3 x e^{x} + 3 x^{2} e^{x} - 9 x e^{x} - 12 e^{x} x + 36 e^{x}}{x^{5}}$

Step 3.10.5.2

Subtract $9 x e^{x}$ from $3 x e^{x}$ .

$f'' (x) = \frac{- 6 x e^{x} + 3 x^{2} e^{x} - 12 e^{x} x + 36 e^{x}}{x^{5}}$

Step 3.10.5.3

Subtract $12 e^{x} x$ from $- 6 x e^{x}$ .

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Step 3.10.5.3.1

Move $e^{x}$ .

$f'' (x) = \frac{- 6 x e^{x} - 12 x e^{x} + 3 x^{2} e^{x} + 36 e^{x}}{x^{5}}$

Step 3.10.5.3.2

Subtract $12 x e^{x}$ from $- 6 x e^{x}$ .

$f'' (x) = \frac{- 18 x e^{x} + 3 x^{2} e^{x} + 36 e^{x}}{x^{5}}$

Step 3.10.6

Reorder terms.

$f'' (x) = \frac{- 18 e^{x} x + 3 e^{x} x^{2} + 36 e^{x}}{x^{5}}$

Step 3.10.7

Simplify the numerator.

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Step 3.10.7.1

Factor $3 e^{x}$ out of $- 18 e^{x} x + 3 e^{x} x^{2} + 36 e^{x}$ .

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Step 3.10.7.1.1

Factor $3 e^{x}$ out of $- 18 e^{x} x$ .

$f'' (x) = \frac{3 e^{x} (- 6 x) + 3 e^{x} x^{2} + 36 e^{x}}{x^{5}}$

Step 3.10.7.1.2

Factor $3 e^{x}$ out of $3 e^{x} x^{2}$ .

$f'' (x) = \frac{3 e^{x} (- 6 x) + 3 e^{x} (x^{2}) + 36 e^{x}}{x^{5}}$

Step 3.10.7.1.3

Factor $3 e^{x}$ out of $36 e^{x}$ .

$f'' (x) = \frac{3 e^{x} (- 6 x) + 3 e^{x} (x^{2}) + 3 e^{x} (12)}{x^{5}}$

Step 3.10.7.1.4

Factor $3 e^{x}$ out of $3 e^{x} (- 6 x) + 3 e^{x} (x^{2})$ .

$f'' (x) = \frac{3 e^{x} (- 6 x + x^{2}) + 3 e^{x} (12)}{x^{5}}$

Step 3.10.7.1.5

Factor $3 e^{x}$ out of $3 e^{x} (- 6 x + x^{2}) + 3 e^{x} (12)$ .

$f'' (x) = \frac{3 e^{x} (- 6 x + x^{2} + 12)}{x^{5}}$

Step 3.10.7.2

Reorder terms.

$f'' (x) = \frac{3 e^{x} (x^{2} - 6 x + 12)}{x^{5}}$

Step 4

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

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

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 $3$ is constant with respect to $x$ , the derivative of $\frac{3 e^{x}}{x^{3}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{e^{x}}{x^{3}}]$ .

$3 \frac{d}{d x} [\frac{e^{x}}{x^{3}}]$

Step 5.1.2

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

Step 5.1.3

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

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Step 5.1.3.1

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

$3 \frac{x^{3} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 5.1.3.2

Multiply $3$ by $2$ .

$3 \frac{x^{3} \frac{d}{d x} [e^{x}] - e^{x} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.1.4

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

$3 \frac{x^{3} e^{x} - e^{x} \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 5.1.5

Differentiate using the Power Rule.

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Step 5.1.5.1

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

$3 \frac{x^{3} e^{x} - e^{x} (3 x^{2})}{x^{6}}$

Step 5.1.5.2

Combine fractions.

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Step 5.1.5.2.1

Multiply $3$ by $- 1$ .

$3 \frac{x^{3} e^{x} - 3 e^{x} x^{2}}{x^{6}}$

Step 5.1.5.2.2

Combine $3$ and $\frac{x^{3} e^{x} - 3 e^{x} x^{2}}{x^{6}}$ .

$\frac{3 (x^{3} e^{x} - 3 e^{x} x^{2})}{x^{6}}$

Step 5.1.6

Simplify.

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Step 5.1.6.1

Apply the distributive property.

$\frac{3 (x^{3} e^{x}) + 3 (- 3 e^{x} x^{2})}{x^{6}}$

Step 5.1.6.2

Simplify the numerator.

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Step 5.1.6.2.1

Multiply $- 3$ by $3$ .

$\frac{3 x^{3} e^{x} - 9 e^{x} x^{2}}{x^{6}}$

Step 5.1.6.2.2

Reorder factors in $3 x^{3} e^{x} - 9 e^{x} x^{2}$ .

$\frac{3 x^{3} e^{x} - 9 x^{2} e^{x}}{x^{6}}$

Step 5.1.6.3

Reorder terms.

$\frac{3 e^{x} x^{3} - 9 e^{x} x^{2}}{x^{6}}$

Step 5.1.6.4

Factor $3 e^{x} x^{2}$ out of $3 e^{x} x^{3} - 9 e^{x} x^{2}$ .

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Step 5.1.6.4.1

Factor $3 e^{x} x^{2}$ out of $3 e^{x} x^{3}$ .

$\frac{3 e^{x} x^{2} (x) - 9 e^{x} x^{2}}{x^{6}}$

Step 5.1.6.4.2

Factor $3 e^{x} x^{2}$ out of $- 9 e^{x} x^{2}$ .

$\frac{3 e^{x} x^{2} (x) + 3 e^{x} x^{2} (- 3)}{x^{6}}$

Step 5.1.6.4.3

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

$\frac{3 e^{x} x^{2} (x - 3)}{x^{6}}$

Step 5.1.6.5

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 5.1.6.5.1

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

$\frac{x^{2} (3 e^{x} (x - 3))}{x^{6}}$

Step 5.1.6.5.2

Cancel the common factors.

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Step 5.1.6.5.2.1

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

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

Step 5.1.6.5.2.2

Cancel the common factor.

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

Step 5.1.6.5.2.3

Rewrite the expression.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$3 e^{x} (x - 3) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

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

$e^{x} = 0$

$x - 3 = 0$

Step 6.3.2

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

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Step 6.3.2.1

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

$e^{x} = 0$

Step 6.3.2.2

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

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Step 6.3.2.2.1

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

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

Step 6.3.2.2.2

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

Undefined

Step 6.3.2.2.3

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

No solution

Step 6.3.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 6.3.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 6.3.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6.3.4

The final solution is all the values that make $3 e^{x} (x - 3) = 0$ true.

$x = 3$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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

$x^{4} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

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

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

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

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

$x = \pm 0$

Step 7.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 8

Critical points to evaluate.

$x = 3$

Step 9

Evaluate the second derivative at $x = 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{3 e^{3} ({(3)}^{2} - 6 \cdot 3 + 12)}{{(3)}^{5}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Cancel the common factor of $3$ and ${(3)}^{5}$ .

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Step 10.1.1

Factor $3$ out of $3 e^{3} ({(3)}^{2} - 6 \cdot 3 + 12)$ .

$\frac{3 (e^{3} ({(3)}^{2} - 6 \cdot 3 + 12))}{{(3)}^{5}}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $3$ out of ${(3)}^{5}$ .

$\frac{3 (e^{3} ({(3)}^{2} - 6 \cdot 3 + 12))}{3 \cdot 3^{4}}$

Step 10.1.2.2

Cancel the common factor.

$\frac{3 (e^{3} ({(3)}^{2} - 6 \cdot 3 + 12))}{3 \cdot 3^{4}}$

Step 10.1.2.3

Rewrite the expression.

$\frac{e^{3} ({(3)}^{2} - 6 \cdot 3 + 12)}{3^{4}}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

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

$\frac{e^{3} (9 - 6 \cdot 3 + 12)}{3^{4}}$

Step 10.2.2

Multiply $- 6$ by $3$ .

$\frac{e^{3} (9 - 18 + 12)}{3^{4}}$

Step 10.2.3

Subtract $18$ from $9$ .

$\frac{e^{3} (- 9 + 12)}{3^{4}}$

Step 10.2.4

Add $- 9$ and $12$ .

$\frac{e^{3} \cdot 3}{3^{4}}$

Step 10.3

Reduce the expression by cancelling the common factors.

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Step 10.3.1

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

$\frac{e^{3} \cdot 3}{81}$

Step 10.3.2

Cancel the common factor of $3$ and $81$ .

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Step 10.3.2.1

Factor $3$ out of $e^{3} \cdot 3$ .

$\frac{3 \cdot e^{3}}{81}$

Step 10.3.2.2

Cancel the common factors.

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Step 10.3.2.2.1

Factor $3$ out of $81$ .

$\frac{3 \cdot e^{3}}{3 \cdot 27}$

Step 10.3.2.2.2

Cancel the common factor.

$\frac{3 \cdot e^{3}}{3 \cdot 27}$

Step 10.3.2.2.3

Rewrite the expression.

$\frac{e^{3}}{27}$

Step 11

$x = 3$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 3$ is a local minimum

Step 12

Find the y-value when $x = 3$ .

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Step 12.1

Replace the variable $x$ with $3$ in the expression.

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

Step 12.2

Simplify the result.

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Step 12.2.1

Cancel the common factor of $3$ and ${(3)}^{3}$ .

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Step 12.2.1.1

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

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

Step 12.2.1.2

Cancel the common factors.

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Step 12.2.1.2.1

Factor $3$ out of ${(3)}^{3}$ .

$f (3) = \frac{3 (e^{3})}{3 \cdot 3^{2}}$

Step 12.2.1.2.2

Cancel the common factor.

$f (3) = \frac{3 e^{3}}{3 \cdot 3^{2}}$

Step 12.2.1.2.3

Rewrite the expression.

$f (3) = \frac{e^{3}}{3^{2}}$

Step 12.2.2

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

$f (3) = \frac{e^{3}}{9}$

Step 12.2.3

The final answer is $\frac{e^{3}}{9}$ .

$y = \frac{e^{3}}{9}$

Step 13

These are the local extrema for $f (x) = \frac{3 e^{x}}{x^{3}}$ .

$(3, \frac{e^{3}}{9})$ is a local minima

Step 14