Calculus Examples

$y = x^{\frac{1}{x}}$

Step 1

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

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

Step 2

Expand the right hand side.

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Step 2.1

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

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

Step 2.2

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

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

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.

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

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $\frac{\ln (x)}{x}$ .

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

Step 3.2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \ln (x)$ and $g (x) = x$ .

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

Step 3.2.3

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

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

Step 3.2.4

Differentiate using the Power Rule.

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Step 3.2.4.1

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

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

Step 3.2.4.2

Cancel the common factor of $x$ .

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Step 3.2.4.2.1

Cancel the common factor.

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

Step 3.2.4.2.2

Rewrite the expression.

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

Step 3.2.4.3

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

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

Step 3.2.4.4

Multiply $- 1$ by $1$ .

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

Step 4

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

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

Step 5

Simplify the right hand side.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $x^{\frac{1}{x}}$ and $x^{2}$ .

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Multiply by $1$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.2.4

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

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

Step 5.3

Apply the distributive property.

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

Step 5.4

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

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

Step 5.5

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

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