Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{x}$ and $\ln^{- 2} (x)$ .

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

Step 4.2

Move $\ln^{- 2} (x)$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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