Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\ln (x)]$

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 2

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

$- \sin (\ln (x)) \frac{1}{x}$

Step 3

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

$- \frac{\sin (\ln (x))}{x}$