Calculus Examples

$f (x) = 3^{\ln (\cos (x))}$

Step 1

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

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Step 1.1

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

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

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $3$ .

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

Step 1.3

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

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

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 $\cos (x)$ .

$3^{\ln (\cos (x))} \ln (3) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\cos (x)])$

Step 2.2

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

$3^{\ln (\cos (x))} \ln (3) (\frac{1}{u_{2}} \frac{d}{d x} [\cos (x)])$

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Reorder the factors of $3^{\ln (\cos (x))} \ln (3) \sec (x) (- \sin (x))$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Separate fractions.

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

Step 5.7

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

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

Step 5.8

Divide $\ln (3) \cdot 3^{\ln (\cos (x))}$ by $1$ .

$- (\ln (3) \cdot 3^{\ln (\cos (x))} \tan (x))$

Step 5.9

Reorder factors in $- \ln (3) \cdot 3^{\ln (\cos (x))} \tan (x)$ .

$- 3^{\ln (\cos (x))} \ln (3) \tan (x)$