Calculus Examples

$y = {(\ln (x))}^{\ln (x)}$

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

$\ln (y) = \ln ({(\ln (x))}^{\ln (x)})$

Step 2

Expand $\ln ({(\ln (x))}^{\ln (x)})$ by moving $\ln (x)$ outside the logarithm.

$\ln (y) = \ln (x) \ln (\ln (x))$

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

$\frac{y'}{y} = \ln (x) \ln (\ln (x))$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $\ln (x) \ln (\ln (x))$ .

$\frac{y'}{y} = \frac{d}{d x} [\ln (x) \ln (\ln (x))]$

Step 3.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \ln (x)$ and $g (x) = \ln (\ln (x))$ .

$\frac{y'}{y} = \ln (x) \frac{d}{d x} [\ln (\ln (x))] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.3

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 3.2.3.1

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

$\frac{y'}{y} = \ln (x) (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\ln (x)]) + \ln (\ln (x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.3.2

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

$\frac{y'}{y} = \ln (x) (\frac{1}{u} \frac{d}{d x} [\ln (x)]) + \ln (\ln (x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.3.3

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

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

Step 3.2.4

Simplify terms.

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Step 3.2.4.1

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

$\frac{y'}{y} = \frac{\ln (x)}{\ln (x)} \frac{d}{d x} [\ln (x)] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.4.2

Cancel the common factor of $\ln (x)$ .

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Step 3.2.4.2.1

Cancel the common factor.

$\frac{y'}{y} = \frac{\ln (x)}{\ln (x)} \frac{d}{d x} [\ln (x)] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.4.2.2

Rewrite the expression.

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

Step 3.2.4.3

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

$\frac{y'}{y} = \frac{d}{d x} [\ln (x)] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)]$

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

$y' = (\frac{1}{x} + \frac{\ln (\ln (x))}{x}) {(\ln (x))}^{\ln (x)}$

Step 5

Simplify the right hand side.

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Step 5.1

Apply the distributive property.

$y' = \frac{1}{x} {\ln (x)}^{\ln (x)} + \frac{\ln (\ln (x))}{x} {\ln (x)}^{\ln (x)}$

Step 5.2

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

$y' = \frac{{\ln (x)}^{\ln (x)}}{x} + \frac{\ln (\ln (x))}{x} {\ln (x)}^{\ln (x)}$

Step 5.3

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

$y' = \frac{{\ln (x)}^{\ln (x)}}{x} + \frac{\ln (\ln (x)) {\ln (x)}^{\ln (x)}}{x}$

Step 5.4

Reorder factors in $\frac{{\ln (x)}^{\ln (x)}}{x} + \frac{\ln (\ln (x)) {\ln (x)}^{\ln (x)}}{x}$ .

$y' = \frac{{\ln (x)}^{\ln (x)}}{x} + \frac{{\ln (x)}^{\ln (x)} \ln (\ln (x))}{x}$