Calculus Examples

$- 4 \sin (x) - 9 \ln (x) + 11$

Step 1

By the Sum Rule, the derivative of $- 4 \sin (x) - 9 \ln (x) + 11$ with respect to $x$ is $\frac{d}{d x} [- 4 \sin (x)] + \frac{d}{d x} [- 9 \ln (x)] + \frac{d}{d x} [11]$ .

$\frac{d}{d x} [- 4 \sin (x)] + \frac{d}{d x} [- 9 \ln (x)] + \frac{d}{d x} [11]$

Step 2

Evaluate $\frac{d}{d x} [- 4 \sin (x)]$ .

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Step 2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \sin (x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\sin (x)]$ .

$- 4 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- 9 \ln (x)] + \frac{d}{d x} [11]$

Step 2.2

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

$- 4 \cos (x) + \frac{d}{d x} [- 9 \ln (x)] + \frac{d}{d x} [11]$

Step 3

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

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Step 3.1

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

$- 4 \cos (x) - 9 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [11]$

Step 3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$- 4 \cos (x) - 9 \frac{1}{x} + \frac{d}{d x} [11]$

Step 3.3

Combine $- 9$ and $\frac{1}{x}$ .

$- 4 \cos (x) + \frac{- 9}{x} + \frac{d}{d x} [11]$

Step 3.4

Move the negative in front of the fraction.

$- 4 \cos (x) - \frac{9}{x} + \frac{d}{d x} [11]$

Step 4

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$- 4 \cos (x) - \frac{9}{x} + 0$

Step 5

Simplify.

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Step 5.1

Add $- 4 \cos (x) - \frac{9}{x}$ and $0$ .

$- 4 \cos (x) - \frac{9}{x}$

Step 5.2

Reorder terms.

$- \frac{9}{x} - 4 \cos (x)$