Calculus Examples

$f (x) = \cos (x) \log (x)$

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 $f (x) = \cos (x)$ and $g (x) = \log (x)$ .

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

Step 2

The derivative of $\log (x)$ with respect to $x$ is $\frac{1}{x \ln (10)}$ .

$\cos (x) \frac{1}{x \ln (10)} + \log (x) \frac{d}{d x} [\cos (x)]$

Step 3

Combine $\cos (x)$ and $\frac{1}{x \ln (10)}$ .

$\frac{\cos (x)}{x \ln (10)} + \log (x) \frac{d}{d x} [\cos (x)]$

Step 4

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

$\frac{\cos (x)}{x \ln (10)} + \log (x) (- \sin (x))$

Step 5

Reorder terms.

$- \sin (x) \log (x) + \frac{\cos (x)}{x \ln (10)}$