Find the Derivative – d/dx 3^(cos(x))

$3^{\cos (x)}$

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 $g (x) = \cos (x)$ .

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

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

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

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

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

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

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

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

Simplify the expression.

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Move $- 1$ to the left of $3^{\cos (x)} \ln (3)$ .

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

Rewrite $- 1 \cdot 3^{\cos (x)}$ as $- 3^{\cos (x)}$ .

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

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

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

Find the Derivative – d/dx 3^(cos(x))

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