Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $e^{x} \sin (x) + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [e^{x} \sin (x)] + \frac{d}{d x} [\cos (x)]$ .

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

Step 2

Evaluate $\frac{d}{d x} [e^{x} \sin (x)]$ .

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Step 2.1

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 $f (x) = e^{x}$ and $g (x) = \sin (x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Reorder terms.

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