Calculus Examples

$\frac{d}{d x} (2 \cos (x) - 3 \ln (x) - 1)$

Step 1

By the Sum Rule, the derivative of $2 \cos (x) - 3 \ln (x) - 1$ with respect to $x$ is $\frac{d}{d x} [2 \cos (x)] + \frac{d}{d x} [- 3 \ln (x)] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [2 \cos (x)] + \frac{d}{d x} [- 3 \ln (x)] + \frac{d}{d x} [- 1]$

Step 2

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

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Step 2.1

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

$2 \frac{d}{d x} [\cos (x)] + \frac{d}{d x} [- 3 \ln (x)] + \frac{d}{d x} [- 1]$

Step 2.2

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

$2 (- \sin (x)) + \frac{d}{d x} [- 3 \ln (x)] + \frac{d}{d x} [- 1]$

Step 2.3

Multiply $- 1$ by $2$ .

$- 2 \sin (x) + \frac{d}{d x} [- 3 \ln (x)] + \frac{d}{d x} [- 1]$

Step 3

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

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Step 3.1

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

$- 2 \sin (x) - 3 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [- 1]$

Step 3.2

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

$- 2 \sin (x) - 3 \frac{1}{x} + \frac{d}{d x} [- 1]$

Step 3.3

Combine $- 3$ and $\frac{1}{x}$ .

$- 2 \sin (x) + \frac{- 3}{x} + \frac{d}{d x} [- 1]$

Step 3.4

Move the negative in front of the fraction.

$- 2 \sin (x) - \frac{3}{x} + \frac{d}{d x} [- 1]$

Step 4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 2 \sin (x) - \frac{3}{x} + 0$

Step 5

Simplify.

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Step 5.1

Add $- 2 \sin (x) - \frac{3}{x}$ and $0$ .

$- 2 \sin (x) - \frac{3}{x}$

Step 5.2

Reorder terms.

$- \frac{3}{x} - 2 \sin (x)$