Calculus Examples

$\sin (x) \ln (x)$

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 $f (x) = \sin (x)$ and $g (x) = \ln (x)$ .

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

Step 2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\sin (x) \frac{1}{x} + \ln (x) \frac{d}{d x} [\sin (x)]$

Step 3

Combine $\sin (x)$ and $\frac{1}{x}$ .

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

Step 4

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

$\frac{\sin (x)}{x} + \ln (x) \cos (x)$

Step 5

Reorder terms.

$\cos (x) \ln (x) + \frac{\sin (x)}{x}$