Calculus Examples

$y = \tan (e^{x})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\tan (e^{x}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $\sec^{2} (e^{x}) e^{x}$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = e^{x} \sec^{2} (e^{x})$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = e^{x} \sec^{2} (e^{x})$