Calculus Examples

$f (x) = e^{\ln (2)} + e^{x} \sin (x)$

Step 1

Exponentiation and log are inverse functions.

$\frac{d}{d x} [2 + e^{x} \sin (x)]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 + e^{x} \sin (x)$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [e^{x} \sin (x)]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [e^{x} \sin (x)]$

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [e^{x} \sin (x)]$

Step 3

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

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Step 3.1

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

$0 + e^{x} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [e^{x}]$

Step 3.2

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

$0 + e^{x} \cos (x) + \sin (x) \frac{d}{d x} [e^{x}]$

Step 3.3

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

$0 + e^{x} \cos (x) + \sin (x) e^{x}$

Step 4

Simplify.

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Step 4.1

Add $0$ and $e^{x} \cos (x) + \sin (x) e^{x}$ .

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

Step 4.2

Reorder terms.

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