Calculus Examples

$2^{x} \sin (x)$

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 $f (x) = 2^{x}$ and $g (x) = \sin (x)$ .

$2^{x} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [2^{x}]$

Step 2

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

$2^{x} \cos (x) + \sin (x) \frac{d}{d x} [2^{x}]$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$2^{x} \cos (x) + \sin (x) (2^{x} \ln (2))$

Step 4

Reorder terms.

$2^{x} \cos (x) + 2^{x} \sin (x) \ln (2)$