Calculus Examples

e^{x} \cos (x)

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

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = e^{x}

$f (x) = e^{x}$ and

g (x) = \cos (x)

$g (x) = \cos (x)$ .

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

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

Step 2

The derivative of

\cos (x)

$\cos (x)$ with respect to

x

$x$ is

- \sin (x)

$- \sin (x)$ .

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

$\frac{d}{d x} [a^{x}]$ is

a^{x} \ln (a)

$a^{x} \ln (a)$ where

a

$a$ =

e

$e$ .

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

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

Step 4

Reorder terms.

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

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