Calculus Examples

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

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

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

Step 2

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 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

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

Step 6

Combine fractions.

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Step 6.1

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

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

Step 6.2

Reorder terms.

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