Calculus Examples

$2 \sin (x) + 3 \cos (x)$

Step 1

By the Sum Rule, the derivative of $2 \sin (x) + 3 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [3 \cos (x)]$ .

$\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [3 \cos (x)]$

Step 2

Evaluate $\frac{d}{d x} [2 \sin (x)]$ .

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin (x)]$ .

$2 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [3 \cos (x)]$

Step 2.2

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

$2 \cos (x) + \frac{d}{d x} [3 \cos (x)]$

Step 3

Evaluate $\frac{d}{d x} [3 \cos (x)]$ .

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Step 3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos (x)$ with respect to $x$ is $3 \frac{d}{d x} [\cos (x)]$ .

$2 \cos (x) + 3 \frac{d}{d x} [\cos (x)]$

Step 3.2

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

$2 \cos (x) + 3 (- \sin (x))$

Step 3.3

Multiply $- 1$ by $3$ .

$2 \cos (x) - 3 \sin (x)$