Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (\cos (x))$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\ln (\cos (x))$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$\frac{1}{\ln (\cos (x))} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\cos (x)])$

Step 2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{1}{\ln (\cos (x))} (\frac{1}{u_{2}} \frac{d}{d x} [\cos (x)])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

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

Step 3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 7.1.2

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

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

Step 7.1.3

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

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

Step 7.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 7.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.4

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\ln (\cos (x))}$ .

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

Step 7.5

Separate fractions.

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

Step 7.6

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

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

Step 7.7

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

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