Calculus Examples

$e^{\tan (x)} - 2$

Step 1

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

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

Step 2

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

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

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Step 2.1.1

To apply the Chain Rule, set $u$ as $\tan (x)$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $\tan (x)$ .

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

Step 2.2

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

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

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$e^{\tan (x)} \sec^{2} (x) + 0$

Step 3.2

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

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