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.

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Step 4.1

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 $g (x) = x^{\ln (x)}$ .

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Use the properties of logarithms to simplify the differentiation.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Raise $\ln (x)$ to the power of $1$ .

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

Step 4.4

Raise $\ln (x)$ to the power of $1$ .

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

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

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 $g (x) = \ln^{2} (x)$ .

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Step 4.7.1

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

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

Step 4.7.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 4.7.3

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

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

Step 4.8

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

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

Step 4.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \ln (x)$ .

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Step 4.9.1

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

$\frac{e^{\ln^{2} (x)}}{x^{\ln (x)}} (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d x} [\ln (x)])$

Step 4.9.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$\frac{e^{\ln^{2} (x)}}{x^{\ln (x)}} (2 u_{3} \frac{d}{d x} [\ln (x)])$

Step 4.9.3

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

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

Step 4.10

Combine fractions.

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Step 4.10.1

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

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

Step 4.10.2

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

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

Step 4.11

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

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

Step 4.12

Multiply $\frac{\ln (x) (2 e^{\ln^{2} (x)})}{x^{\ln (x)}}$ by $\frac{1}{x}$ .

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

Step 4.13

Multiply $x^{\ln (x)}$ by $x$ .

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Step 4.13.1

Raise $x$ to the power of $1$ .

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

Step 4.13.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.14

Simplify the numerator.

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Step 4.14.1

Rewrite using the commutative property of multiplication.

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

Step 4.14.2

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

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

Step 4.14.3

Reorder factors in $\ln (x^{2}) e^{\ln^{2} (x)}$ .

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

Step 5

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

$y' = \frac{e^{\ln^{2} (x)} \ln (x^{2})}{x^{\ln (x) + 1}}$

Step 6

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

$\frac{d y}{d x} = \frac{e^{\ln^{2} (x)} \ln (x^{2})}{x^{\ln (x) + 1}}$