Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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 $g (x) = x$ .

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

Step 1.2

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

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

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

Step 1.3.2

Multiply $- 1$ by $1$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

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

Step 1.4.2

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

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Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Factor $e^{x}$ out of $e^{x} (x) + e^{x} \cdot - 1$ .

$f' (x) = \frac{e^{x} (x - 1)}{x^{2}}$

Step 2

Find the second derivative.

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Step 2.1

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $2$ .

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

Step 2.3

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 $g (x) = x - 1$ .

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

Step 2.4

Differentiate.

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Step 2.4.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 2.4.2

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

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

Step 2.4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Add $1$ and $0$ .

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

Step 2.4.4.2

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

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

Step 2.5

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

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

Step 2.6

Differentiate using the Power Rule.

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Step 2.6.1

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

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

Step 2.6.2

Simplify with factoring out.

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Step 2.6.2.1

Multiply $2$ by $- 1$ .

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

Step 2.6.2.2

Factor $x$ out of $x^{2} (e^{x} + (x - 1) e^{x}) - 2 e^{x} (x - 1) x$ .

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Step 2.6.2.2.1

Factor $x$ out of $x^{2} (e^{x} + (x - 1) e^{x})$ .

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

Step 2.6.2.2.2

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

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

Step 2.6.2.2.3

Factor $x$ out of $x (x (e^{x} + (x - 1) e^{x})) + x (- 2 e^{x} (x - 1))$ .

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

Step 2.7

Cancel the common factors.

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Step 2.7.1

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

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

Step 2.7.2

Cancel the common factor.

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

Step 2.7.3

Rewrite the expression.

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Apply the distributive property.

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

Step 2.8.4

Simplify the numerator.

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Step 2.8.4.1

Combine the opposite terms in $x e^{x} + x (x e^{x}) + x (- 1 e^{x}) - 2 e^{x} x - 2 e^{x} \cdot - 1$ .

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Step 2.8.4.1.1

Reorder the factors in the terms $x e^{x}$ and $x (- 1 e^{x})$ .

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

Step 2.8.4.1.2

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

$f'' (x) = \frac{x (x e^{x}) + 0 - 2 e^{x} x - 2 e^{x} \cdot - 1}{x^{3}}$

Step 2.8.4.1.3

Add $x (x e^{x})$ and $0$ .

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

Step 2.8.4.2

Simplify each term.

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Step 2.8.4.2.1

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

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Step 2.8.4.2.1.1

Move $x$ .

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

Step 2.8.4.2.1.2

Multiply $x$ by $x$ .

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

Step 2.8.4.2.2

Multiply $- 1$ by $- 2$ .

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

Step 2.8.4.3

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

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

Step 2.8.5

Reorder terms.

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

Step 2.8.6

Reorder factors in $\frac{e^{x} x^{2} - 2 e^{x} x + 2 e^{x}}{x^{3}}$ .

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

Step 3

Find the third derivative.

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Step 3.1

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

Step 3.2

Differentiate using the Sum Rule.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

Multiply $3$ by $2$ .

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

Step 3.2.2

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

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

Step 3.3

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 $g (x) = e^{x}$ .

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

Step 3.4

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

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

Step 3.5

Differentiate.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Differentiate.

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Step 3.8.1

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

$f''' (x) = \frac{x^{3} (x^{2} e^{x} + e^{x} (2 x) - 2 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} (2 e^{x})) - (x^{2} e^{x} - 2 x e^{x} + 2 e^{x}) \frac{d}{d x} x^{3}}{x^{6}}$

Step 3.8.2

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

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

Step 3.8.3

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

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

Step 3.9

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

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

Step 3.10

Differentiate using the Power Rule.

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Step 3.10.1

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

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

Step 3.10.2

Simplify with factoring out.

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Step 3.10.2.1

Multiply $3$ by $- 1$ .

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

Step 3.10.2.2

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

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Step 3.10.2.2.1

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

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

Step 3.10.2.2.2

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

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

Step 3.10.2.2.3

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

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

Step 3.11

Cancel the common factors.

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Step 3.11.1

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

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

Step 3.11.2

Cancel the common factor.

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

Step 3.11.3

Rewrite the expression.

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

Step 3.12

Simplify.

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Step 3.12.1

Apply the distributive property.

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

Step 3.12.2

Apply the distributive property.

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

Step 3.12.3

Apply the distributive property.

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

Step 3.12.4

Simplify the numerator.

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Step 3.12.4.1

Combine the opposite terms in $x (x^{2} e^{x}) + x (e^{x} (2 x)) + x (- 2 (x e^{x})) + x (- 2 e^{x}) + x (2 e^{x}) - 3 (x^{2} e^{x}) - 3 (- 2 x e^{x}) - 3 (2 e^{x})$ .

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Step 3.12.4.1.1

Reorder the factors in the terms $x (e^{x} (2 x))$ and $x (- 2 (x e^{x}))$ .

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

Step 3.12.4.1.2

Subtract $2 e^{x} x \cdot x$ from $2 e^{x} x \cdot x$ .

$f''' (x) = \frac{x (x^{2} e^{x}) + 0 + x (- 2 e^{x}) + x (2 e^{x}) - 3 (x^{2} e^{x}) - 3 (- 2 x e^{x}) - 3 (2 e^{x})}{x^{4}}$

Step 3.12.4.1.3

Add $x (x^{2} e^{x})$ and $0$ .

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

Step 3.12.4.1.4

Reorder the factors in the terms $x (- 2 e^{x})$ and $x (2 e^{x})$ .

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

Step 3.12.4.1.5

Add $- 2 e^{x} x$ and $2 e^{x} x$ .

$f''' (x) = \frac{x (x^{2} e^{x}) + 0 - 3 (x^{2} e^{x}) - 3 (- 2 x e^{x}) - 3 (2 e^{x})}{x^{4}}$

Step 3.12.4.1.6

Add $x (x^{2} e^{x})$ and $0$ .

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

Step 3.12.4.2

Simplify each term.

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Step 3.12.4.2.1

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

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Step 3.12.4.2.1.1

Move $x^{2}$ .

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

Step 3.12.4.2.1.2

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

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Step 3.12.4.2.1.2.1

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

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

Step 3.12.4.2.1.2.2

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

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

Step 3.12.4.2.1.3

Add $2$ and $1$ .

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

Step 3.12.4.2.2

Multiply $- 2$ by $- 3$ .

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

Step 3.12.4.2.3

Multiply $2$ by $- 3$ .

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

Step 3.12.5

Reorder terms.

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

Step 3.12.6

Reorder factors in $\frac{e^{x} x^{3} - 3 e^{x} x^{2} + 6 e^{x} x - 6 e^{x}}{x^{4}}$ .

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

Step 4

Find the fourth derivative.

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Step 4.1

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

Step 4.2

Differentiate using the Sum Rule.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

Multiply $4$ by $2$ .

$f^{4} (x) = \frac{x^{4} \frac{d}{d x} (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.2.2

By the Sum Rule, the derivative of $x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{3} e^{x}] + \frac{d}{d x} [- 3 x^{2} e^{x}] + \frac{d}{d x} [6 x e^{x}] + \frac{d}{d x} [- 6 e^{x}]$ .

$f^{4} (x) = \frac{x^{4} (\frac{d}{d x} (x^{3} e^{x}) + \frac{d}{d x} (- 3 x^{2} e^{x}) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.3

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

$f^{4} (x) = \frac{x^{4} (x^{3} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 3 x^{2} e^{x}) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.4

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 3 x^{2} e^{x}) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5

Differentiate.

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Step 4.5.1

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) + \frac{d}{d x} (- 3 x^{2} e^{x}) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.5.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2} e^{x}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2} e^{x}]$ .

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 \frac{d}{d x} (x^{2} e^{x}) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.6

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 $g (x) = e^{x}$ .

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.7

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} \frac{d}{d x} (x^{2})) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.8

Differentiate.

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Step 4.8.1

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + \frac{d}{d x} (6 x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.8.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x e^{x}$ with respect to $x$ is $6 \frac{d}{d x} [x e^{x}]$ .

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 \frac{d}{d x} (x e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.9

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.10

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x} \frac{d}{d x} (x)) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.11

Differentiate.

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Step 4.11.1

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x} \cdot 1) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.11.2

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) + \frac{d}{d x} (- 6 e^{x})) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.11.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 e^{x}$ with respect to $x$ is $- 6 \frac{d}{d x} [e^{x}]$ .

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 \frac{d}{d x} e^{x}) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.12

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x}) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) \frac{d}{d x} x^{4}}{x^{8}}$

Step 4.13

Differentiate using the Power Rule.

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Step 4.13.1

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

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x}) - (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) (4 x^{3})}{x^{8}}$

Step 4.13.2

Simplify with factoring out.

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Step 4.13.2.1

Multiply $4$ by $- 1$ .

$f^{4} (x) = \frac{x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x}) - 4 (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) x^{3}}{x^{8}}$

Step 4.13.2.2

Factor $x^{3}$ out of $x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x}) - 4 (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) x^{3}$ .

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Step 4.13.2.2.1

Factor $x^{3}$ out of $x^{4} (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x})$ .

$f^{4} (x) = \frac{x^{3} (x (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x})) - 4 (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) x^{3}}{x^{8}}$

Step 4.13.2.2.2

Factor $x^{3}$ out of $- 4 (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}) x^{3}$ .

$f^{4} (x) = \frac{x^{3} (x (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x})) + x^{3} (- 4 (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}))}{x^{8}}$

Step 4.13.2.2.3

Factor $x^{3}$ out of $x^{3} (x (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x})) + x^{3} (- 4 (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}))$ .

$f^{4} (x) = \frac{x^{3} (x (x^{3} e^{x} + e^{x} (3 x^{2}) - 3 (x^{2} e^{x} + e^{x} (2 x)) + 6 (x e^{x} + e^{x}) - 6 e^{x}) - 4 (x^{3} e^{x} - 3 x^{2} e^{x} + 6 x e^{x} - 6 e^{x}))}{x^{8}}$

Step 4.14

Cancel the common factors.

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Step 4.14.1

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

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

Step 4.14.2

Cancel the common factor.

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

Step 4.14.3

Rewrite the expression.

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

Step 4.15

Simplify.

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Step 4.15.1

Apply the distributive property.

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

Step 4.15.2

Apply the distributive property.

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

Step 4.15.3

Apply the distributive property.

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

Step 4.15.4

Apply the distributive property.

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

Step 4.15.5

Simplify the numerator.

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Step 4.15.5.1

Combine the opposite terms in $x (x^{3} e^{x}) + x (e^{x} (3 x^{2})) + x (- 3 (x^{2} e^{x})) + x (- 3 (e^{x} (2 x))) + x (6 (x e^{x})) + x (6 e^{x}) + x (- 6 e^{x}) - 4 (x^{3} e^{x}) - 4 (- 3 x^{2} e^{x}) - 4 (6 x e^{x}) - 4 (- 6 e^{x})$ .

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Step 4.15.5.1.1

Reorder the factors in the terms $x (e^{x} (3 x^{2}))$ and $x (- 3 (x^{2} e^{x}))$ .

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

Step 4.15.5.1.2

Subtract $3 e^{x} x^{2} x$ from $3 e^{x} x^{2} x$ .

$f^{4} (x) = \frac{x (x^{3} e^{x}) + 0 + x (- 3 (e^{x} (2 x))) + x (6 (x e^{x})) + x (6 e^{x}) + x (- 6 e^{x}) - 4 (x^{3} e^{x}) - 4 (- 3 x^{2} e^{x}) - 4 (6 x e^{x}) - 4 (- 6 e^{x})}{x^{5}}$

Step 4.15.5.1.3

Add $x (x^{3} e^{x})$ and $0$ .

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

Step 4.15.5.1.4

Reorder the factors in the terms $x (6 e^{x})$ and $x (- 6 e^{x})$ .

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

Step 4.15.5.1.5

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

$f^{4} (x) = \frac{x (x^{3} e^{x}) + x (- 3 (e^{x} (2 x))) + x (6 (x e^{x})) + 0 - 4 (x^{3} e^{x}) - 4 (- 3 x^{2} e^{x}) - 4 (6 x e^{x}) - 4 (- 6 e^{x})}{x^{5}}$

Step 4.15.5.1.6

Add $x (x^{3} e^{x}) + x (- 3 (e^{x} (2 x))) + x (6 (x e^{x}))$ and $0$ .

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

Step 4.15.5.2

Simplify each term.

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Step 4.15.5.2.1

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

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Step 4.15.5.2.1.1

Move $x^{3}$ .

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

Step 4.15.5.2.1.2

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

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Step 4.15.5.2.1.2.1

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

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

Step 4.15.5.2.1.2.2

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

$f^{4} (x) = \frac{x^{3 + 1} e^{x} + x (- 3 (e^{x} (2 x))) + x (6 (x e^{x})) - 4 (x^{3} e^{x}) - 4 (- 3 x^{2} e^{x}) - 4 (6 x e^{x}) - 4 (- 6 e^{x})}{x^{5}}$

Step 4.15.5.2.1.3

Add $3$ and $1$ .

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

Step 4.15.5.2.2

Rewrite using the commutative property of multiplication.

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

Step 4.15.5.2.3

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

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Step 4.15.5.2.3.1

Move $x$ .

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

Step 4.15.5.2.3.2

Multiply $x$ by $x$ .

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

Step 4.15.5.2.4

Multiply $- 3$ by $2$ .

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

Step 4.15.5.2.5

Rewrite using the commutative property of multiplication.

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

Step 4.15.5.2.6

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

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Step 4.15.5.2.6.1

Move $x$ .

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

Step 4.15.5.2.6.2

Multiply $x$ by $x$ .

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

Step 4.15.5.2.7

Multiply $- 3$ by $- 4$ .

$f^{4} (x) = \frac{x^{4} e^{x} - 6 x^{2} e^{x} + 6 x^{2} e^{x} - 4 x^{3} e^{x} + 12 (x^{2} e^{x}) - 4 (6 x e^{x}) - 4 (- 6 e^{x})}{x^{5}}$

Step 4.15.5.2.8

Multiply $6$ by $- 4$ .

$f^{4} (x) = \frac{x^{4} e^{x} - 6 x^{2} e^{x} + 6 x^{2} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 (x e^{x}) - 4 (- 6 e^{x})}{x^{5}}$

Step 4.15.5.2.9

Multiply $- 6$ by $- 4$ .

$f^{4} (x) = \frac{x^{4} e^{x} - 6 x^{2} e^{x} + 6 x^{2} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}}{x^{5}}$

Step 4.15.5.3

Combine the opposite terms in $x^{4} e^{x} - 6 x^{2} e^{x} + 6 x^{2} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}$ .

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Step 4.15.5.3.1

Add $- 6 x^{2} e^{x}$ and $6 x^{2} e^{x}$ .

$f^{4} (x) = \frac{x^{4} e^{x} + 0 - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}}{x^{5}}$

Step 4.15.5.3.2

Add $x^{4} e^{x}$ and $0$ .

$f^{4} (x) = \frac{x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}}{x^{5}}$

Step 4.15.6

Reorder terms.

$f^{4} (x) = \frac{e^{x} x^{4} - 4 e^{x} x^{3} + 12 e^{x} x^{2} - 24 e^{x} x + 24 e^{x}}{x^{5}}$

Step 4.15.7

Reorder factors in $\frac{e^{x} x^{4} - 4 e^{x} x^{3} + 12 e^{x} x^{2} - 24 e^{x} x + 24 e^{x}}{x^{5}}$ .

$f^{4} (x) = \frac{x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}}{x^{5}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $\frac{x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}}{x^{5}}$ .

$\frac{x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}}{x^{5}}$