Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 \tan (x)$ .

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $2 \tan (x)$ .

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

Step 2

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \tan (x)$ with respect to $x$ is $2 \frac{d}{d x} [\tan (x)]$ .

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

Step 2.2

Move $2$ to the left of $e^{2 \tan (x)}$ .

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

Step 3

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

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