Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

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

Step 1.3

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

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

Step 2

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

$\cos (e^{x}) e^{x}$

Step 3

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

$e^{x} \cos (e^{x})$