Calculus Examples

$\tan (e^{x}) + e^{\tan (x)}$

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [\tan (e^{x})]$ .

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Step 2.1

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

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $e^{x}$ .

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

Step 2.1.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$\sec^{2} (u_{1}) \frac{d}{d x} [e^{x}] + \frac{d}{d x} [e^{\tan (x)}]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $e^{x}$ .

$\sec^{2} (e^{x}) \frac{d}{d x} [e^{x}] + \frac{d}{d x} [e^{\tan (x)}]$

Step 2.2

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

$\sec^{2} (e^{x}) e^{x} + \frac{d}{d x} [e^{\tan (x)}]$

Step 3

Evaluate $\frac{d}{d x} [e^{\tan (x)}]$ .

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u_{2}$ as $\tan (x)$ .

$\sec^{2} (e^{x}) e^{x} + \frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [\tan (x)]$

Step 3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$\sec^{2} (e^{x}) e^{x} + e^{u_{2}} \frac{d}{d x} [\tan (x)]$

Step 3.1.3

Replace all occurrences of $u_{2}$ with $\tan (x)$ .

$\sec^{2} (e^{x}) e^{x} + e^{\tan (x)} \frac{d}{d x} [\tan (x)]$

Step 3.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (e^{x}) e^{x} + e^{\tan (x)} \sec^{2} (x)$

Step 4

Reorder terms.

$e^{x} \sec^{2} (e^{x}) + e^{\tan (x)} \sec^{2} (x)$