Calculus Examples

$5 e^{x} - 9 \sin (x)$

Step 1

By the Sum Rule, the derivative of $5 e^{x} - 9 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [5 e^{x}] + \frac{d}{d x} [- 9 \sin (x)]$ .

$\frac{d}{d x} [5 e^{x}] + \frac{d}{d x} [- 9 \sin (x)]$

Step 2

Evaluate $\frac{d}{d x} [5 e^{x}]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{x}$ with respect to $x$ is $5 \frac{d}{d x} [e^{x}]$ .

$5 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 9 \sin (x)]$

Step 2.2

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

$5 e^{x} + \frac{d}{d x} [- 9 \sin (x)]$

Step 3

Evaluate $\frac{d}{d x} [- 9 \sin (x)]$ .

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Step 3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 \sin (x)$ with respect to $x$ is $- 9 \frac{d}{d x} [\sin (x)]$ .

$5 e^{x} - 9 \frac{d}{d x} [\sin (x)]$

Step 3.2

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

$5 e^{x} - 9 \cos (x)$