Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = \cos (x)$ .

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

Step 2

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

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

Step 3

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

$e^{x} (- \sin (x)) + \cos (x) e^{x}$

Step 4

Reorder terms.

$- e^{x} \sin (x) + e^{x} \cos (x)$