Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [\frac{1}{x}]$ .

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Step 2.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [- \ln (x)]$

Step 2.2

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

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

Step 3

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

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (x)$ with respect to $x$ is $- \frac{d}{d x} [\ln (x)]$ .

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

Step 3.2

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

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

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5

Reorder terms.

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