Use Logarithmic Differentiation to Find the Derivative

$y = x^{\sqrt{x}}$

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

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

Step 2

Expand the right hand side.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 2.2

Expand $\ln (x^{x^{\frac{1}{2}}})$ by moving $x^{\frac{1}{2}}$ outside the logarithm.

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

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

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

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

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

$\frac{y'}{y} = \frac{d}{d x} [x^{\frac{1}{2}} \ln (x)]$

Step 3.2.2

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = \ln (x)$ .

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

Step 3.2.3

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

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

Step 3.2.4

Combine fractions.

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Step 3.2.4.1

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

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

Step 3.2.4.2

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 3.2.5

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.2.5.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 3.2.5.1.1

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

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

Step 3.2.5.1.2

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

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

Step 3.2.5.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.5.3

Combine the numerators over the common denominator.

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

Step 3.2.5.4

Subtract $1$ from $2$ .

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

Step 3.2.6

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

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

Step 3.2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.2.8

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

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

Step 3.2.9

Combine the numerators over the common denominator.

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

Step 3.2.10

Simplify the numerator.

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Step 3.2.10.1

Multiply $- 1$ by $2$ .

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

Step 3.2.10.2

Subtract $2$ from $1$ .

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

Step 3.2.11

Move the negative in front of the fraction.

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

Step 3.2.12

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

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

Step 3.2.13

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

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

Step 3.2.14

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

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

Step 5

Simplify the right hand side.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Simplify each term.

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Step 5.4.1

Factor $x^{\frac{1}{2}}$ out of $x^{\sqrt{x}}$ .

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

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Multiply by $1$ .

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

Step 5.4.2.2

Cancel the common factor.

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

Step 5.4.2.3

Rewrite the expression.

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

Step 5.4.2.4

Divide $x^{\sqrt{x} - \frac{1}{2}}$ by $1$ .

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

Step 5.4.3

Factor $x^{\frac{1}{2}}$ out of $\ln (x) x^{\sqrt{x}}$ .

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

Step 5.4.4

Cancel the common factors.

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Step 5.4.4.1

Factor $x^{\frac{1}{2}}$ out of $2 x^{\frac{1}{2}}$ .

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

Step 5.4.4.2

Cancel the common factor.

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

Step 5.4.4.3

Rewrite the expression.

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

Step 5.4.5

Simplify the numerator.

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Step 5.4.5.1

To write $\sqrt{x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.4.5.2

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

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

Step 5.4.5.3

Combine the numerators over the common denominator.

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

Step 5.4.5.4

Move $2$ to the left of $\sqrt{x}$ .

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

Step 5.5

To write $x^{\sqrt{x} - \frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.6

Combine $x^{\sqrt{x} - \frac{1}{2}}$ and $\frac{2}{2}$ .

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

Step 5.7

Combine the numerators over the common denominator.

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

Step 5.8

Simplify the numerator.

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Step 5.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 5.8.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 5.8.3

Move $2$ to the left of $x^{x^{\frac{1}{2}} - \frac{1}{2}}$ .

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

Step 5.8.4

Simplify each term.

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Step 5.8.4.1

To write $x^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.8.4.2

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

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

Step 5.8.4.3

Combine the numerators over the common denominator.

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

Step 5.8.4.4

Move $2$ to the left of $x^{\frac{1}{2}}$ .

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

Step 5.8.5

Factor $x^{\frac{2 x^{\frac{1}{2}} - 1}{2}}$ out of $2 x^{\frac{2 x^{\frac{1}{2}} - 1}{2}} + \ln (x) x^{\frac{2 x^{\frac{1}{2}} - 1}{2}}$ .

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Step 5.8.5.1

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

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

Step 5.8.5.2

Factor $x^{\frac{2 x^{\frac{1}{2}} - 1}{2}}$ out of $2 x^{\frac{2 x^{\frac{1}{2}} - 1}{2}}$ .

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

Step 5.8.5.3

Factor $x^{\frac{2 x^{\frac{1}{2}} - 1}{2}}$ out of $x^{\frac{2 x^{\frac{1}{2}} - 1}{2}} \ln (x)$ .

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

Step 5.8.5.4

Factor $x^{\frac{2 x^{\frac{1}{2}} - 1}{2}}$ out of $x^{\frac{2 x^{\frac{1}{2}} - 1}{2}} \cdot 2 + x^{\frac{2 x^{\frac{1}{2}} - 1}{2}} (\ln (x))$ .

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

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