Calculus Examples

$\ln (\ln (\ln (\ln (x))))$

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\ln (\ln (\ln (x)))$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\ln (\ln (\ln (x)))$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $\ln (\ln (x))$ .

$\frac{1}{\ln (\ln (\ln (x)))} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\ln (\ln (x))])$

Step 2.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\frac{1}{\ln (\ln (\ln (x)))} (\frac{1}{u_{2}} \frac{d}{d x} [\ln (\ln (x))])$

Step 2.3

Replace all occurrences of $u_{2}$ with $\ln (\ln (x))$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

To apply the Chain Rule, set $u_{3}$ as $\ln (x)$ .

$\frac{1}{\ln (\ln (x)) \ln (\ln (\ln (x)))} (\frac{d}{d u_{3}} [\ln (u_{3})] \frac{d}{d x} [\ln (x)])$

Step 4.2

The derivative of $\ln (u_{3})$ with respect to $u_{3}$ is $\frac{1}{u_{3}}$ .

$\frac{1}{\ln (\ln (x)) \ln (\ln (\ln (x)))} (\frac{1}{u_{3}} \frac{d}{d x} [\ln (x)])$

Step 4.3

Replace all occurrences of $u_{3}$ with $\ln (x)$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Combine fractions.

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Step 7.1

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

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

Step 7.2

Reorder terms.

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