Calculus Examples

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

Step 1

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 $g (x) = e^{x} + \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $e^{x} + \ln (x)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $e^{x} + \ln (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\frac{1}{e^{x} + \ln (x)} (e^{x} + \frac{1}{x})$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $\frac{1}{e^{x} + \ln (x)} (e^{x} + \frac{1}{x})$ .

$(e^{x} + \frac{1}{x}) \frac{1}{e^{x} + \ln (x)}$

Step 5.2

Multiply $e^{x} + \frac{1}{x}$ by $\frac{1}{e^{x} + \ln (x)}$ .

$\frac{e^{x} + \frac{1}{x}}{e^{x} + \ln (x)}$

Step 5.3

Multiply the numerator and denominator of the fraction by $x$ .

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Step 5.3.1

Multiply $\frac{e^{x} + \frac{1}{x}}{e^{x} + \ln (x)}$ by $\frac{x}{x}$ .

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

Step 5.3.2

Combine.

$\frac{x (e^{x} + \frac{1}{x})}{x (e^{x} + \ln (x))}$

Step 5.4

Apply the distributive property.

$\frac{x e^{x} + x \frac{1}{x}}{x e^{x} + x \ln (x)}$

Step 5.5

Cancel the common factor of $x$ .

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Step 5.5.1

Cancel the common factor.

$\frac{x e^{x} + x \frac{1}{x}}{x e^{x} + x \ln (x)}$

Step 5.5.2

Rewrite the expression.

$\frac{x e^{x} + 1}{x e^{x} + x \ln (x)}$

Step 5.6

Factor $x$ out of $x e^{x} + x \ln (x)$ .

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Step 5.6.1

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

$\frac{x e^{x} + 1}{x (e^{x}) + x \ln (x)}$

Step 5.6.2

Factor $x$ out of $x \ln (x)$ .

$\frac{x e^{x} + 1}{x (e^{x}) + x (\ln (x))}$

Step 5.6.3

Factor $x$ out of $x (e^{x}) + x (\ln (x))$ .

$\frac{x e^{x} + 1}{x (e^{x} + \ln (x))}$