Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite $x^{\frac{1}{x}}$ as $e^{\ln (x^{\frac{1}{x}})}$ .

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

Step 1.2

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

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

Step 2

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

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

Step 3

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 $g (x) = \frac{\ln (x)}{x}$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $\frac{\ln (x)}{x}$ .

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

Step 4

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 $g (x) = x$ .

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

Step 5

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

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

Step 6

Differentiate using the Power Rule.

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Step 6.1

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

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

Step 6.2

Cancel the common factor of $x$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

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

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

Step 6.4

Combine fractions.

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Step 6.4.1

Multiply $- 1$ by $1$ .

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

Step 6.4.2

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

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply $e^{\frac{\ln (x)}{x}}$ by $1$ .

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

Step 7.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.3

Reorder terms.

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