Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Use the properties of logarithms to simplify the differentiation.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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 3.6.1

To apply the Chain Rule, set $u_{1}$ as $\ln^{2} (x)$ .

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

Step 3.6.2

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

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

Step 3.6.3

Replace all occurrences of $u_{1}$ with $\ln^{2} (x)$ .

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

Step 3.7

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

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 3.7.3

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

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

Step 3.8

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

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

Step 3.9

Combine fractions.

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Step 3.9.1

Combine $\frac{1}{x}$ and $2$ .

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

Step 3.9.2

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

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

Step 3.9.3

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

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

Step 3.10

Simplify the numerator.

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Step 3.10.1

Rewrite using the commutative property of multiplication.

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

Step 3.10.2

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

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Step 3.10.2.1

Reorder $\ln (x)$ and $2$ .

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

Step 3.10.2.2

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

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

Step 3.10.3

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

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

Step 4

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

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

Step 5

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

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