Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{1}{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{1}{x}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{x}$ .

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

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{x}$ .

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

Step 3

Simplify the expression.

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Step 3.1

Multiply $x$ by $1$ .

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

Step 3.2

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

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

Step 4

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

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

Step 5

Raise $x$ to the power of $1$ .

$- (x^{1} x^{- 2})$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{1 - 2}$

Step 7

Subtract $2$ from $1$ .

$- x^{- 1}$

Step 8

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

$- \frac{1}{x}$