Calculus Examples

$\ln (x) - 2 x^{2}$

Step 1

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

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

Step 2

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

$\frac{1}{x} + \frac{d}{d x} [- 2 x^{2}]$

Step 3

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

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Step 3.1

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

$\frac{1}{x} - 2 \frac{d}{d x} [x^{2}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{x} - 2 (2 x)$

Step 3.3

Multiply $2$ by $- 2$ .

$\frac{1}{x} - 4 x$

Step 4

Reorder terms.

$- 4 x + \frac{1}{x}$