Calculus Examples

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

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

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

Step 2

The derivative of $\sinh (x)$ with respect to $x$ is $\cosh (x)$ .

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

Step 3

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

$e^{x} \cosh (x) + \sinh (x) e^{x}$

Step 4

Reorder terms.

$e^{x} \cosh (x) + e^{x} \sinh (x)$