Calculus Examples

$y = \ln (\cos (e^{x}))$

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 $g (x) = \cos (e^{x})$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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 $g (x) = e^{x}$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $e^{x}$ .

$\sec (e^{x}) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [e^{x}])$

Step 3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$\sec (e^{x}) (- \sin (u_{2}) \frac{d}{d x} [e^{x}])$

Step 3.3

Replace all occurrences of $u_{2}$ with $e^{x}$ .

$\sec (e^{x}) (- \sin (e^{x}) \frac{d}{d x} [e^{x}])$

Step 4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\sec (e^{x}) (- \sin (e^{x}) e^{x})$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $\sec (e^{x}) (- \sin (e^{x}) e^{x})$ .

$- e^{x} \sec (e^{x}) \sin (e^{x})$

Step 5.2

Rewrite $\sec (e^{x})$ in terms of sines and cosines.

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

Step 5.3

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

$- \frac{e^{x}}{\cos (e^{x})} \sin (e^{x})$

Step 5.4

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

$- \frac{\sin (e^{x}) e^{x}}{\cos (e^{x})}$

Step 5.5

Separate fractions.

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

Step 5.6

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

$- (\frac{e^{x}}{1} \tan (e^{x}))$

Step 5.7

Divide $e^{x}$ by $1$ .

$- e^{x} \tan (e^{x})$