Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

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

Step 3.3

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

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $x$ .

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

Step 3.4

Simplify terms.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

Cancel the common factor of $x^{\ln (x)}$ and $x$ .

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Step 3.4.3.1

Factor $x$ out of $\ln (x) x^{\ln (x)}$ .

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

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

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

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

Step 3.4.3.2.2

Factor $x$ out of $x^{1}$ .

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

Step 3.4.3.2.3

Cancel the common factor.

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

Step 3.4.3.2.4

Rewrite the expression.

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

Step 3.4.3.2.5

Divide $\ln (x) x^{\ln (x) - 1}$ by $1$ .

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

Step 3.5

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$x' (\ln (x) x^{\ln (x) - 1}) + \ln (x) x^{\ln (x) - 1} x'$

Step 3.6

Reorder the factors of $x' (\ln (x) x^{\ln (x) - 1})$ .

$x^{\ln (x) - 1} \ln (x) x' + \ln (x) x^{\ln (x) - 1} x'$

Step 3.7

Add $x^{\ln (x) - 1} \ln (x) x'$ and $\ln (x) x^{\ln (x) - 1} x'$ .

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Step 3.7.1

Reorder $\ln (x)$ and $x^{\ln (x) - 1}$ .

$x^{\ln (x) - 1} \ln (x) x' + x^{\ln (x) - 1} \ln (x) x'$

Step 3.7.2

Add $x^{\ln (x) - 1} \ln (x) x'$ and $x^{\ln (x) - 1} \ln (x) x'$ .

$2 x^{\ln (x) - 1} \ln (x) x'$

Step 4

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

$1 = 2 x^{\ln (x) - 1} \ln (x) x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $2 x^{\ln (x) - 1} \ln (x) x' = 1$ .

$2 x^{\ln (x) - 1} \ln (x) x' = 1$

Step 5.2

Divide each term in $2 x^{\ln (x) - 1} \ln (x) x' = 1$ by $2 x^{\ln (x) - 1} \ln (x)$ and simplify.

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Step 5.2.1

Divide each term in $2 x^{\ln (x) - 1} \ln (x) x' = 1$ by $2 x^{\ln (x) - 1} \ln (x)$ .

$\frac{2 x^{\ln (x) - 1} \ln (x) x'}{2 x^{\ln (x) - 1} \ln (x)} = \frac{1}{2 x^{\ln (x) - 1} \ln (x)}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x^{\ln (x) - 1} \ln (x) x'}{2 x^{\ln (x) - 1} \ln (x)} = \frac{1}{2 x^{\ln (x) - 1} \ln (x)}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{x^{\ln (x) - 1} \ln (x) x'}{x^{\ln (x) - 1} \ln (x)} = \frac{1}{2 x^{\ln (x) - 1} \ln (x)}$

Step 5.2.2.2

Cancel the common factor of $x^{\ln (x) - 1}$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{x^{\ln (x) - 1} \ln (x) x'}{x^{\ln (x) - 1} \ln (x)} = \frac{1}{2 x^{\ln (x) - 1} \ln (x)}$

Step 5.2.2.2.2

Rewrite the expression.

$\frac{\ln (x) x'}{\ln (x)} = \frac{1}{2 x^{\ln (x) - 1} \ln (x)}$

Step 5.2.2.3

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

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Step 5.2.2.3.1

Cancel the common factor.

$\frac{\ln (x) x'}{\ln (x)} = \frac{1}{2 x^{\ln (x) - 1} \ln (x)}$

Step 5.2.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{1}{2 x^{\ln (x) - 1} \ln (x)}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Reorder factors in $\frac{1}{2 x^{\ln (x) - 1} \ln (x)}$ .

$x' = \frac{1}{2 \ln (x) x^{\ln (x) - 1}}$

Step 6

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

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