Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x \ln (\frac{1}{x})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x \ln (\frac{1}{x})$ .

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with $x \ln (\frac{1}{x})$ .

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

Step 3

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

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{1}{x}$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $\frac{1}{x}$ .

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

Step 4.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

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

Step 4.3

Replace all occurrences of $u_{2}$ with $\frac{1}{x}$ .

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

Step 5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{x}$ .

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

Step 6

Multiply $x$ by $1$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify the expression.

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Step 10.1

Add $1$ and $1$ .

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

Step 10.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 11

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

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

Step 12

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

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Step 12.1

Move $x^{- 2}$ .

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

Step 12.2

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

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

Step 12.3

Add $- 2$ and $2$ .

$e^{x \ln (\frac{1}{x})} (x^{0} \cdot - 1 + \ln (\frac{1}{x}) \frac{d}{d x} [x])$

Step 13

Simplify $x^{0} \cdot - 1$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify.

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Step 16.1

Apply the distributive property.

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

Step 16.2

Combine terms.

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Step 16.2.1

Move $- 1$ to the left of $e^{x \ln (\frac{1}{x})}$ .

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

Step 16.2.2

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

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

Step 16.3

Reorder terms.

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