Calculus Examples

$e^{x} \ln (x^{8})$

Step 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 $f (x) = e^{x}$ and $g (x) = \ln (x^{8})$ .

$e^{x} \frac{d}{d x} [\ln (x^{8})] + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{8}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{8}$ .

$e^{x} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{8}]) + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$e^{x} (\frac{1}{u} \frac{d}{d x} [x^{8}]) + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 2.3

Replace all occurrences of $u$ with $x^{8}$ .

$e^{x} (\frac{1}{x^{8}} \frac{d}{d x} [x^{8}]) + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Combine $\frac{1}{x^{8}}$ and $e^{x}$ .

$\frac{e^{x}}{x^{8}} \frac{d}{d x} [x^{8}] + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$\frac{e^{x}}{x^{8}} (8 x^{7}) + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3.3

Simplify terms.

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Step 3.3.1

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

$\frac{8 e^{x}}{x^{8}} x^{7} + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3.3.2

Combine $\frac{8 e^{x}}{x^{8}}$ and $x^{7}$ .

$\frac{8 e^{x} x^{7}}{x^{8}} + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3.3.3

Cancel the common factor of $x^{7}$ and $x^{8}$ .

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Step 3.3.3.1

Factor $x^{7}$ out of $8 e^{x} x^{7}$ .

$\frac{x^{7} (8 e^{x})}{x^{8}} + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $x^{7}$ out of $x^{8}$ .

$\frac{x^{7} (8 e^{x})}{x^{7} x} + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{x^{7} (8 e^{x})}{x^{7} x} + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{8 e^{x}}{x} + \ln (x^{8}) \frac{d}{d x} [e^{x}]$

Step 4

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

$\frac{8 e^{x}}{x} + \ln (x^{8}) e^{x}$

Step 5

Reorder terms.

$e^{x} \ln (x^{8}) + \frac{8 e^{x}}{x}$