Calculus Examples

\ln (\ln (x))

$\ln (\ln (x))$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = \ln (x)

$f (x) = \ln (x)$ and

g (x) = \ln (x)

$g (x) = \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set

u

$u$ as

\ln (x)

$\ln (x)$ .

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

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

Step 1.2

The derivative of

\ln (u)

$\ln (u)$ with respect to

u

$u$ is

\frac{1}{u}

$\frac{1}{u}$ .

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

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

Step 1.3

Replace all occurrences of

u

$u$ with

\ln (x)

$\ln (x)$ .

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

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

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

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

Step 2

The derivative of

\ln (x)

$\ln (x)$ with respect to

x

$x$ is

\frac{1}{x}

$\frac{1}{x}$ .

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

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

Step 3

Combine fractions.

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Step 3.1

Multiply

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

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

\frac{1}{x}

$\frac{1}{x}$ .

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

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

Step 3.2

Reorder terms.

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

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

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

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

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$