Calculus Examples

$10^{\sin (x)}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 10^{x}$ and $g (x) = \sin (x)$ .

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Step 1.1

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

$\frac{d}{d u} [10^{u}] \frac{d}{d x} [\sin (x)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $10$ .

$10^{u} \ln (10) \frac{d}{d x} [\sin (x)]$

Step 1.3

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

$10^{\sin (x)} \ln (10) \frac{d}{d x} [\sin (x)]$

Step 2

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

$10^{\sin (x)} \ln (10) \cos (x)$

Step 3

Reorder the factors of $10^{\sin (x)} \ln (10) \cos (x)$ .

$10^{\sin (x)} \cos (x) \ln (10)$