Calculus Examples

$f (x) = 4 - 2 \ln (x)$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $4 - 2 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 2 \ln (x)]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- 2 \ln (x)]$

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 \ln (x)]$

Step 2

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

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Step 2.1

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

$0 - 2 \frac{d}{d x} [\ln (x)]$

Step 2.2

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

$0 - 2 \frac{1}{x}$

Step 2.3

Combine $- 2$ and $\frac{1}{x}$ .

$0 + \frac{- 2}{x}$

Step 2.4

Move the negative in front of the fraction.

$0 - \frac{2}{x}$

Step 3

Subtract $\frac{2}{x}$ from $0$ .

$- \frac{2}{x}$