Calculus Examples

$f (x) = x^{7} e^{x}$

Step 1

Find the first derivative.

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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 $f (x) = x^{7}$ and $g (x) = e^{x}$ .

$f' (x) = x^{7} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{7})$

Step 1.2

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

$f' (x) = x^{7} e^{x} + e^{x} \frac{d}{d x} (x^{7})$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$f' (x) = x^{7} e^{x} + e^{x} (7 x^{6})$

Step 1.4

Simplify.

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Step 1.4.1

Reorder terms.

$f' (x) = e^{x} x^{7} + 7 e^{x} x^{6}$

Step 1.4.2

Reorder factors in $e^{x} x^{7} + 7 e^{x} x^{6}$ .

$f' (x) = x^{7} e^{x} + 7 x^{6} e^{x}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $x^{7} e^{x} + 7 x^{6} e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{7} e^{x}] + \frac{d}{d x} [7 x^{6} e^{x}]$ .

$f'' (x) = \frac{d}{d x} (x^{7} e^{x}) + \frac{d}{d x} (7 x^{6} e^{x})$

Step 2.2

Evaluate $\frac{d}{d x} [x^{7} e^{x}]$ .

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Step 2.2.1

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 $f (x) = x^{7}$ and $g (x) = e^{x}$ .

$f'' (x) = x^{7} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{7}) + \frac{d}{d x} (7 x^{6} e^{x})$

Step 2.2.2

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

$f'' (x) = x^{7} e^{x} + e^{x} \frac{d}{d x} (x^{7}) + \frac{d}{d x} (7 x^{6} e^{x})$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$f'' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + \frac{d}{d x} (7 x^{6} e^{x})$

Step 2.3

Evaluate $\frac{d}{d x} [7 x^{6} e^{x}]$ .

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Step 2.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{6} e^{x}$ with respect to $x$ is $7 \frac{d}{d x} [x^{6} e^{x}]$ .

$f'' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 7 \frac{d}{d x} (x^{6} e^{x})$

Step 2.3.2

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 $f (x) = x^{6}$ and $g (x) = e^{x}$ .

$f'' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 7 (x^{6} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{6}))$

Step 2.3.3

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

$f'' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 7 (x^{6} e^{x} + e^{x} \frac{d}{d x} (x^{6}))$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 7 (x^{6} e^{x} + e^{x} (6 x^{5}))$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$f'' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 7 (x^{6} e^{x}) + 7 (e^{x} (6 x^{5}))$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $6$ by $7$ .

$f'' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 7 x^{6} e^{x} + 42 (e^{x} (x^{5}))$

Step 2.4.2.2

Add $e^{x} (7 x^{6})$ and $7 x^{6} e^{x}$ .

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Step 2.4.2.2.1

Move $e^{x}$ .

$f'' (x) = x^{7} e^{x} + 7 x^{6} e^{x} + 7 x^{6} e^{x} + 42 e^{x} x^{5}$

Step 2.4.2.2.2

Add $7 x^{6} e^{x}$ and $7 x^{6} e^{x}$ .

$f'' (x) = x^{7} e^{x} + 14 x^{6} e^{x} + 42 e^{x} x^{5}$

Step 2.4.3

Reorder terms.

$f'' (x) = e^{x} x^{7} + 14 e^{x} x^{6} + 42 e^{x} x^{5}$

Step 2.4.4

Reorder factors in $e^{x} x^{7} + 14 e^{x} x^{6} + 42 e^{x} x^{5}$ .

$f'' (x) = x^{7} e^{x} + 14 x^{6} e^{x} + 42 x^{5} e^{x}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $x^{7} e^{x} + 14 x^{6} e^{x} + 42 x^{5} e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{7} e^{x}] + \frac{d}{d x} [14 x^{6} e^{x}] + \frac{d}{d x} [42 x^{5} e^{x}]$ .

$f''' (x) = \frac{d}{d x} (x^{7} e^{x}) + \frac{d}{d x} (14 x^{6} e^{x}) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.2

Evaluate $\frac{d}{d x} [x^{7} e^{x}]$ .

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Step 3.2.1

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 $f (x) = x^{7}$ and $g (x) = e^{x}$ .

$f''' (x) = x^{7} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{7}) + \frac{d}{d x} (14 x^{6} e^{x}) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.2.2

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

$f''' (x) = x^{7} e^{x} + e^{x} \frac{d}{d x} (x^{7}) + \frac{d}{d x} (14 x^{6} e^{x}) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + \frac{d}{d x} (14 x^{6} e^{x}) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.3

Evaluate $\frac{d}{d x} [14 x^{6} e^{x}]$ .

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Step 3.3.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x^{6} e^{x}$ with respect to $x$ is $14 \frac{d}{d x} [x^{6} e^{x}]$ .

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 \frac{d}{d x} (x^{6} e^{x}) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.3.2

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 $f (x) = x^{6}$ and $g (x) = e^{x}$ .

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.3.3

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

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x} + e^{x} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x} + e^{x} (6 x^{5})) + \frac{d}{d x} (42 x^{5} e^{x})$

Step 3.4

Evaluate $\frac{d}{d x} [42 x^{5} e^{x}]$ .

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Step 3.4.1

Since $42$ is constant with respect to $x$ , the derivative of $42 x^{5} e^{x}$ with respect to $x$ is $42 \frac{d}{d x} [x^{5} e^{x}]$ .

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x} + e^{x} (6 x^{5})) + 42 \frac{d}{d x} (x^{5} e^{x})$

Step 3.4.2

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 $f (x) = x^{5}$ and $g (x) = e^{x}$ .

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x} + e^{x} (6 x^{5})) + 42 (x^{5} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{5}))$

Step 3.4.3

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

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x} + e^{x} (6 x^{5})) + 42 (x^{5} e^{x} + e^{x} \frac{d}{d x} (x^{5}))$

Step 3.4.4

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

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x} + e^{x} (6 x^{5})) + 42 (x^{5} e^{x} + e^{x} (5 x^{4}))$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x}) + 14 (e^{x} (6 x^{5})) + 42 (x^{5} e^{x} + e^{x} (5 x^{4}))$

Step 3.5.2

Apply the distributive property.

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 (x^{6} e^{x}) + 14 (e^{x} (6 x^{5})) + 42 (x^{5} e^{x}) + 42 (e^{x} (5 x^{4}))$

Step 3.5.3

Combine terms.

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Step 3.5.3.1

Multiply $6$ by $14$ .

$f''' (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 14 x^{6} e^{x} + 84 (e^{x} (x^{5})) + 42 (x^{5} e^{x}) + 42 (e^{x} (5 x^{4}))$

Step 3.5.3.2

Add $e^{x} (7 x^{6})$ and $14 x^{6} e^{x}$ .

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Step 3.5.3.2.1

Move $e^{x}$ .

$f''' (x) = x^{7} e^{x} + 7 x^{6} e^{x} + 14 x^{6} e^{x} + 84 e^{x} x^{5} + 42 (x^{5} e^{x}) + 42 (e^{x} (5 x^{4}))$

Step 3.5.3.2.2

Add $7 x^{6} e^{x}$ and $14 x^{6} e^{x}$ .

$f''' (x) = x^{7} e^{x} + 21 x^{6} e^{x} + 84 e^{x} x^{5} + 42 (x^{5} e^{x}) + 42 (e^{x} (5 x^{4}))$

Step 3.5.3.3

Multiply $5$ by $42$ .

$f''' (x) = x^{7} e^{x} + 21 x^{6} e^{x} + 84 e^{x} x^{5} + 42 x^{5} e^{x} + 210 (e^{x} (x^{4}))$

Step 3.5.3.4

Add $84 e^{x} x^{5}$ and $42 x^{5} e^{x}$ .

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Step 3.5.3.4.1

Move $e^{x}$ .

$f''' (x) = x^{7} e^{x} + 21 x^{6} e^{x} + 84 x^{5} e^{x} + 42 x^{5} e^{x} + 210 e^{x} x^{4}$

Step 3.5.3.4.2

Add $84 x^{5} e^{x}$ and $42 x^{5} e^{x}$ .

$f''' (x) = x^{7} e^{x} + 21 x^{6} e^{x} + 126 x^{5} e^{x} + 210 e^{x} x^{4}$

Step 3.5.4

Reorder terms.

$f''' (x) = e^{x} x^{7} + 21 e^{x} x^{6} + 126 e^{x} x^{5} + 210 e^{x} x^{4}$

Step 3.5.5

Reorder factors in $e^{x} x^{7} + 21 e^{x} x^{6} + 126 e^{x} x^{5} + 210 e^{x} x^{4}$ .

$f''' (x) = x^{7} e^{x} + 21 x^{6} e^{x} + 126 x^{5} e^{x} + 210 x^{4} e^{x}$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $x^{7} e^{x} + 21 x^{6} e^{x} + 126 x^{5} e^{x} + 210 x^{4} e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{7} e^{x}] + \frac{d}{d x} [21 x^{6} e^{x}] + \frac{d}{d x} [126 x^{5} e^{x}] + \frac{d}{d x} [210 x^{4} e^{x}]$ .

$f^{4} (x) = \frac{d}{d x} (x^{7} e^{x}) + \frac{d}{d x} (21 x^{6} e^{x}) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.2

Evaluate $\frac{d}{d x} [x^{7} e^{x}]$ .

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Step 4.2.1

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 $f (x) = x^{7}$ and $g (x) = e^{x}$ .

$f^{4} (x) = x^{7} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{7}) + \frac{d}{d x} (21 x^{6} e^{x}) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.2.2

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

$f^{4} (x) = x^{7} e^{x} + e^{x} \frac{d}{d x} (x^{7}) + \frac{d}{d x} (21 x^{6} e^{x}) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + \frac{d}{d x} (21 x^{6} e^{x}) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.3

Evaluate $\frac{d}{d x} [21 x^{6} e^{x}]$ .

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Step 4.3.1

Since $21$ is constant with respect to $x$ , the derivative of $21 x^{6} e^{x}$ with respect to $x$ is $21 \frac{d}{d x} [x^{6} e^{x}]$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 \frac{d}{d x} (x^{6} e^{x}) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.3.2

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 $f (x) = x^{6}$ and $g (x) = e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.3.3

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

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + \frac{d}{d x} (126 x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.4

Evaluate $\frac{d}{d x} [126 x^{5} e^{x}]$ .

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Step 4.4.1

Since $126$ is constant with respect to $x$ , the derivative of $126 x^{5} e^{x}$ with respect to $x$ is $126 \frac{d}{d x} [x^{5} e^{x}]$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 \frac{d}{d x} (x^{5} e^{x}) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.4.2

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 $f (x) = x^{5}$ and $g (x) = e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 (x^{5} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{5})) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.4.3

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

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 (x^{5} e^{x} + e^{x} \frac{d}{d x} (x^{5})) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.4.4

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

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 (x^{5} e^{x} + e^{x} (5 x^{4})) + \frac{d}{d x} (210 x^{4} e^{x})$

Step 4.5

Evaluate $\frac{d}{d x} [210 x^{4} e^{x}]$ .

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Step 4.5.1

Since $210$ is constant with respect to $x$ , the derivative of $210 x^{4} e^{x}$ with respect to $x$ is $210 \frac{d}{d x} [x^{4} e^{x}]$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 (x^{5} e^{x} + e^{x} (5 x^{4})) + 210 \frac{d}{d x} (x^{4} e^{x})$

Step 4.5.2

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

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 (x^{5} e^{x} + e^{x} (5 x^{4})) + 210 (x^{4} \frac{d}{d x} (e^{x}) + e^{x} \frac{d}{d x} (x^{4}))$

Step 4.5.3

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

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 (x^{5} e^{x} + e^{x} (5 x^{4})) + 210 (x^{4} e^{x} + e^{x} \frac{d}{d x} (x^{4}))$

Step 4.5.4

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

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x} + e^{x} (6 x^{5})) + 126 (x^{5} e^{x} + e^{x} (5 x^{4})) + 210 (x^{4} e^{x} + e^{x} (4 x^{3}))$

Step 4.6

Simplify.

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Step 4.6.1

Apply the distributive property.

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x}) + 21 (e^{x} (6 x^{5})) + 126 (x^{5} e^{x} + e^{x} (5 x^{4})) + 210 (x^{4} e^{x} + e^{x} (4 x^{3}))$

Step 4.6.2

Apply the distributive property.

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x}) + 21 (e^{x} (6 x^{5})) + 126 (x^{5} e^{x}) + 126 (e^{x} (5 x^{4})) + 210 (x^{4} e^{x} + e^{x} (4 x^{3}))$

Step 4.6.3

Apply the distributive property.

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 (x^{6} e^{x}) + 21 (e^{x} (6 x^{5})) + 126 (x^{5} e^{x}) + 126 (e^{x} (5 x^{4})) + 210 (x^{4} e^{x}) + 210 (e^{x} (4 x^{3}))$

Step 4.6.4

Combine terms.

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Step 4.6.4.1

Multiply $6$ by $21$ .

$f^{4} (x) = x^{7} e^{x} + e^{x} (7 x^{6}) + 21 x^{6} e^{x} + 126 (e^{x} (x^{5})) + 126 (x^{5} e^{x}) + 126 (e^{x} (5 x^{4})) + 210 (x^{4} e^{x}) + 210 (e^{x} (4 x^{3}))$

Step 4.6.4.2

Add $e^{x} (7 x^{6})$ and $21 x^{6} e^{x}$ .

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Step 4.6.4.2.1

Move $e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + 7 x^{6} e^{x} + 21 x^{6} e^{x} + 126 e^{x} x^{5} + 126 (x^{5} e^{x}) + 126 (e^{x} (5 x^{4})) + 210 (x^{4} e^{x}) + 210 (e^{x} (4 x^{3}))$

Step 4.6.4.2.2

Add $7 x^{6} e^{x}$ and $21 x^{6} e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 126 e^{x} x^{5} + 126 (x^{5} e^{x}) + 126 (e^{x} (5 x^{4})) + 210 (x^{4} e^{x}) + 210 (e^{x} (4 x^{3}))$

Step 4.6.4.3

Multiply $5$ by $126$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 126 e^{x} x^{5} + 126 x^{5} e^{x} + 630 (e^{x} (x^{4})) + 210 (x^{4} e^{x}) + 210 (e^{x} (4 x^{3}))$

Step 4.6.4.4

Add $126 e^{x} x^{5}$ and $126 x^{5} e^{x}$ .

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Step 4.6.4.4.1

Move $e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 126 x^{5} e^{x} + 126 x^{5} e^{x} + 630 e^{x} x^{4} + 210 (x^{4} e^{x}) + 210 (e^{x} (4 x^{3}))$

Step 4.6.4.4.2

Add $126 x^{5} e^{x}$ and $126 x^{5} e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 252 x^{5} e^{x} + 630 e^{x} x^{4} + 210 (x^{4} e^{x}) + 210 (e^{x} (4 x^{3}))$

Step 4.6.4.5

Multiply $4$ by $210$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 252 x^{5} e^{x} + 630 e^{x} x^{4} + 210 x^{4} e^{x} + 840 (e^{x} (x^{3}))$

Step 4.6.4.6

Add $630 e^{x} x^{4}$ and $210 x^{4} e^{x}$ .

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Step 4.6.4.6.1

Move $e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 252 x^{5} e^{x} + 630 x^{4} e^{x} + 210 x^{4} e^{x} + 840 e^{x} x^{3}$

Step 4.6.4.6.2

Add $630 x^{4} e^{x}$ and $210 x^{4} e^{x}$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 252 x^{5} e^{x} + 840 x^{4} e^{x} + 840 e^{x} x^{3}$

Step 4.6.5

Reorder terms.

$f^{4} (x) = e^{x} x^{7} + 28 e^{x} x^{6} + 252 e^{x} x^{5} + 840 e^{x} x^{4} + 840 e^{x} x^{3}$

Step 4.6.6

Reorder factors in $e^{x} x^{7} + 28 e^{x} x^{6} + 252 e^{x} x^{5} + 840 e^{x} x^{4} + 840 e^{x} x^{3}$ .

$f^{4} (x) = x^{7} e^{x} + 28 x^{6} e^{x} + 252 x^{5} e^{x} + 840 x^{4} e^{x} + 840 x^{3} e^{x}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $x^{7} e^{x} + 28 x^{6} e^{x} + 252 x^{5} e^{x} + 840 x^{4} e^{x} + 840 x^{3} e^{x}$ .

$x^{7} e^{x} + 28 x^{6} e^{x} + 252 x^{5} e^{x} + 840 x^{4} e^{x} + 840 x^{3} e^{x}$