Calculus Examples

$f (x) = \ln (x) + \frac{1}{x}$

Step 1

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

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

Step 2

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

$\frac{1}{x} + \frac{d}{d x} [\frac{1}{x}]$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

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