Calculus Examples

y = x^{ln (x)}

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

Step 1

Let

y = f (x)

$y = f (x)$ , take the natural logarithm of both sides

ln (y) = ln (f (x))

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

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

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

Step 2

Expand the right hand side.

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Step 2.1

Expand

ln (x^{ln (x)})

$\ln (x^{\ln (x)})$ by moving

ln (x)

$\ln (x)$ outside the logarithm.

ln (y) = ln (x) ln (x)

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

Step 2.2

Raise

ln (x)

$\ln (x)$ to the power of

1

$1$ .

ln (y) = {ln}^{1} (x) ln (x)

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

Step 2.3

Raise

ln (x)

$\ln (x)$ to the power of

1

$1$ .

ln (y) = {ln}^{1} (x) {ln}^{1} (x)

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

Step 2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

ln (y) = {ln (x)}^{1 + 1}

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

Step 2.5

Add

1

$1$ and

1

$1$ .

ln (y) = {ln}^{2} (x)

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

ln (y) = {ln}^{2} (x)

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

Step 3

Differentiate the expression using the chain rule, keeping in mind that

y

$y$ is a function of

x

$x$ .

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Step 3.1

Differentiate the left hand side

ln (y)

$\ln (y)$ using the chain rule.

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

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate

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

$\frac{y'}{y} = \frac{d}{d x} [\ln^{2} (x)]$

Step 3.2.2

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = x^{2}$ and

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

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Step 3.2.2.1

To apply the Chain Rule, set

$u$ as

$\ln (x)$ .

$\frac{y'}{y} = \frac{d}{d u} [u^{2}] \frac{d}{d x} [\ln (x)]$

Step 3.2.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

$n = 2$ .

$\frac{y'}{y} = 2 u \frac{d}{d x} [\ln (x)]$

Step 3.2.2.3

Replace all occurrences of

$u$ with

$\ln (x)$ .

$\frac{y'}{y} = 2 \ln (x) \frac{d}{d x} [\ln (x)]$

Step 3.2.3

The derivative of

$\ln (x)$ with respect to

$x$ is

$\frac{1}{x}$ .

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

Step 3.2.4

Combine fractions.

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Step 3.2.4.1

Combine

$\frac{1}{x}$ and

$2$ .

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

Step 3.2.4.2

Combine

$\frac{2}{x}$ and

$\ln (x)$ .

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

Step 3.2.5

Simplify

$2 \ln (x)$ by moving

$2$ inside the logarithm.

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

Step 4

Isolate

$y'$ and substitute the original function for

$y$ in the right hand side.

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

Step 5

Simplify the right hand side.

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Step 5.1

Combine

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

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

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

Step 5.2

Cancel the common factor of

$x^{\ln (x)}$ and

$x$ .

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Step 5.2.1

Factor

$x$ out of

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

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Raise

$x$ to the power of

$1$ .

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

Step 5.2.2.2

Factor

$x$ out of

$x^{1}$ .

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

Step 5.2.2.3

Cancel the common factor.

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

Step 5.2.2.4

Rewrite the expression.

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

Step 5.2.2.5

Divide

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

$1$ .

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

Step 5.3

Reorder factors in

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

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

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