Calculus Examples

$f (x) = 2 x + \ln (x)$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [\ln (x)]$

Step 2.3

Multiply $2$ by $1$ .

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

Step 3

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

$2 + \frac{1}{x}$

Step 4

Reorder terms.

$\frac{1}{x} + 2$