Calculus Examples

$- 2 x^{2} + 16 \ln (x)$

Step 1

By the Sum Rule, the derivative of $- 2 x^{2} + 16 \ln (x)$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [16 \ln (x)]$ .

$\frac{d}{d x} [- 2 x^{2}] + \frac{d}{d x} [16 \ln (x)]$

Step 2

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

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Step 2.1

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

$- 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [16 \ln (x)]$

Step 2.2

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

$- 2 (2 x) + \frac{d}{d x} [16 \ln (x)]$

Step 2.3

Multiply $2$ by $- 2$ .

$- 4 x + \frac{d}{d x} [16 \ln (x)]$

Step 3

Evaluate $\frac{d}{d x} [16 \ln (x)]$ .

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Step 3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 \ln (x)$ with respect to $x$ is $16 \frac{d}{d x} [\ln (x)]$ .

$- 4 x + 16 \frac{d}{d x} [\ln (x)]$

Step 3.2

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

$- 4 x + 16 \frac{1}{x}$

Step 3.3

Combine $16$ and $\frac{1}{x}$ .

$- 4 x + \frac{16}{x}$