Calculus Examples

$\log_{3} (\cos (x))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{3} (x)$ and $g (x) = \cos (x)$ .

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Step 1.1

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

$\frac{d}{d u} [\log_{3} (u)] \frac{d}{d x} [\cos (x)]$

Step 1.2

The derivative of $\log_{3} (u)$ with respect to $u$ is $\frac{1}{u \ln (3)}$ .

$\frac{1}{u \ln (3)} \frac{d}{d x} [\cos (x)]$

Step 1.3

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

$\frac{1}{\cos (x) \ln (3)} \frac{d}{d x} [\cos (x)]$

Step 2

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

$\frac{1}{\cos (x) \ln (3)} (- \sin (x))$

Step 3

Combine $\frac{1}{\cos (x) \ln (3)}$ and $\sin (x)$ .

$- \frac{\sin (x)}{\cos (x) \ln (3)}$

Step 4

Simplify.

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Step 4.1

Separate fractions.

$- (\frac{1}{\ln (3)} \cdot \frac{\sin (x)}{\cos (x)})$

Step 4.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$- (\frac{1}{\ln (3)} \tan (x))$

Step 4.3

Combine $\frac{1}{\ln (3)}$ and $\tan (x)$ .

$- \frac{\tan (x)}{\ln (3)}$