Calculus Examples

$f (x) = \log (x) \cdot \sin (x)$

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

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

Step 2

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

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

Step 3

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

$\log (x) \cos (x) + \sin (x) \frac{1}{x \ln (10)}$

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Reorder terms.

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