Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{\sin (x)}$ and $g (x) = \cos (x)$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

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

$e^{\sin (x)} (- \sin (x)) + \cos (x) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [\sin (x)])$

Step 3.2

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

$e^{\sin (x)} (- \sin (x)) + \cos (x) (e^{u} \frac{d}{d x} [\sin (x)])$

Step 3.3

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

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

Step 4

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

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

Step 5

Raise $\cos (x)$ to the power of $1$ .

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

Step 6

Raise $\cos (x)$ to the power of $1$ .

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

Step 7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8

Add $1$ and $1$ .

$e^{\sin (x)} (- \sin (x)) + \cos^{2} (x) e^{\sin (x)}$

Step 9

Reorder terms.

$- e^{\sin (x)} \sin (x) + e^{\sin (x)} \cos^{2} (x)$