Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Differentiate both sides of the equation.

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

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

Tap for more steps...

Step 4.1

Use the properties of logarithms to simplify the differentiation.

Tap for more steps...

Step 4.1.1

Rewrite ${(\ln (x))}^{\ln (x)}$ as $e^{\ln ({(\ln (x))}^{\ln (x)})}$ .

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

Step 4.1.2

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

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 4.2.3

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

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

Step 4.3

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

$e^{\ln (x) \ln (\ln (x))} (\ln (x) \frac{d}{d x} [\ln (\ln (x))] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)])$

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

$e^{\ln (x) \ln (\ln (x))} (\ln (x) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\ln (x)]) + \ln (\ln (x)) \frac{d}{d x} [\ln (x)])$

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.5

Simplify terms.

Tap for more steps...

Step 4.5.1

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

$e^{\ln (x) \ln (\ln (x))} (\frac{\ln (x)}{\ln (x)} \frac{d}{d x} [\ln (x)] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)])$

Step 4.5.2

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

Tap for more steps...

Step 4.5.2.1

Cancel the common factor.

$e^{\ln (x) \ln (\ln (x))} (\frac{\ln (x)}{\ln (x)} \frac{d}{d x} [\ln (x)] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)])$

Step 4.5.2.2

Rewrite the expression.

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

Step 4.5.3

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

$e^{\ln (x) \ln (\ln (x))} (\frac{d}{d x} [\ln (x)] + \ln (\ln (x)) \frac{d}{d x} [\ln (x)])$

Step 4.6

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

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

Step 4.7

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

$e^{\ln (x) \ln (\ln (x))} (\frac{1}{x} + \ln (\ln (x)) \frac{1}{x})$

Step 4.8

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

$e^{\ln (x) \ln (\ln (x))} (\frac{1}{x} + \frac{\ln (\ln (x))}{x})$

Step 4.9

Simplify.

Tap for more steps...

Step 4.9.1

Apply the distributive property.

$e^{\ln (x) \ln (\ln (x))} \frac{1}{x} + e^{\ln (x) \ln (\ln (x))} \frac{\ln (\ln (x))}{x}$

Step 4.9.2

Combine terms.

Tap for more steps...

Step 4.9.2.1

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

$\frac{e^{\ln (x) \ln (\ln (x))}}{x} + e^{\ln (x) \ln (\ln (x))} \frac{\ln (\ln (x))}{x}$

Step 4.9.2.2

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

$\frac{e^{\ln (x) \ln (\ln (x))}}{x} + \frac{e^{\ln (x) \ln (\ln (x))} \ln (\ln (x))}{x}$

Step 5

Reform the equation by setting the left side equal to the right side.

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

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

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