Calculus Examples

$y = x^{\frac{2}{x}}$

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Use the properties of logarithms to simplify the differentiation.

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Step 3.1.1

Rewrite $x^{\frac{2}{x}}$ as $e^{\ln (x^{\frac{2}{x}})}$ .

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

Step 3.1.2

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

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

Step 3.2

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

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

Step 3.3

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 $g (x) = \frac{2 \ln (x)}{x}$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $\frac{2 \ln (x)}{x}$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $\frac{2 \ln (x)}{x}$ .

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

Step 3.4

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 \ln (x)}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{\ln (x)}{x}]$ .

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

Step 3.5

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 $g (x) = x$ .

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

Step 3.6

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

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

Step 3.7

Differentiate using the Power Rule.

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Step 3.7.1

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

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

Step 3.7.2

Cancel the common factor of $x$ .

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Step 3.7.2.1

Cancel the common factor.

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

Step 3.7.2.2

Rewrite the expression.

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

Step 3.7.3

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

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

Step 3.7.4

Combine fractions.

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Step 3.7.4.1

Multiply $- 1$ by $1$ .

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

Step 3.7.4.2

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

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

Step 3.7.4.3

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

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Simplify the numerator.

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Step 3.8.3.1

Simplify each term.

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Step 3.8.3.1.1

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

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

Step 3.8.3.1.2

Multiply $2$ by $1$ .

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

Step 3.8.3.1.3

Move $2$ to the left of $e^{\frac{\ln (x^{2})}{x}}$ .

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

Step 3.8.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.8.3.1.5

Multiply $2$ by $- 1$ .

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

Step 3.8.3.1.6

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

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

Step 3.8.3.1.7

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

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Step 3.8.3.1.7.1

Reorder $\ln (x)$ and $- 2$ .

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

Step 3.8.3.1.7.2

Simplify $- 2 \cdot \ln (x)$ by moving $2$ inside the logarithm.

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

Step 3.8.3.2

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

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

Step 3.8.4

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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