Calculus Examples

${(x y)}^{x} = e$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Use the properties of logarithms to simplify the differentiation.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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 $g (x) = x \ln (x y)$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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 $g (x) = \ln (x y)$ .

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

Step 2.4

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 $g (x) = x y$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{2}$ as $x y$ .

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

Step 2.4.2

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

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

Step 2.4.3

Replace all occurrences of $u_{2}$ with $x y$ .

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

Step 2.5

Simplify terms.

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Step 2.5.1

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

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

Step 2.5.2

Cancel the common factor of $x$ .

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Step 2.5.2.1

Cancel the common factor.

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

Step 2.5.2.2

Rewrite the expression.

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

Step 2.6

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 $g (x) = y$ .

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

Step 2.7

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 2.8

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

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

Step 2.9

Multiply $y$ by $1$ .

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

Step 2.10

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

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

Step 2.11

Multiply $\ln (x y)$ by $1$ .

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

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

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

Step 2.12.2

Apply the distributive property.

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

Step 2.12.3

Combine terms.

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Step 2.12.3.1

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

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

Step 2.12.3.2

Combine $\frac{x}{y}$ and $y'$ .

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

Step 2.12.3.3

Combine $e^{x \ln (x y)}$ and $\frac{x y'}{y}$ .

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

Step 2.12.3.4

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

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

Step 2.12.3.5

Cancel the common factor of $y$ .

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Step 2.12.3.5.1

Cancel the common factor.

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

Step 2.12.3.5.2

Rewrite the expression.

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

Step 2.12.3.6

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

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

Step 2.12.4

Reorder terms.

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

Step 3

Since $e$ is constant with respect to $x$ , the derivative of $e$ with respect to $x$ is $0$ .

$0$

Step 4

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

$e^{x \ln (x y)} \ln (x y) + \frac{e^{x \ln (x y)} (x y')}{y} + e^{x \ln (x y)} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify the left side.

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Step 5.1.1

Reorder factors in $e^{x \ln (x y)} \ln (x y) + \frac{e^{x \ln (x y)} x y'}{y} + e^{x \ln (x y)}$ .

$e^{x \ln (x y)} \ln (x y) + \frac{x y' e^{x \ln (x y)}}{y} + e^{x \ln (x y)} = 0$

Step 5.1.2

Reorder $x$ and $y'$ .

$e^{x \ln (x y)} \ln (x y) + \frac{y' x e^{x \ln (x y)}}{y} + e^{x \ln (x y)} = 0$

Step 5.1.3

Reorder $e^{x \ln (x y)} \ln (x y)$ and $\frac{y' x e^{x \ln (x y)}}{y}$ .

$\frac{y' x e^{x \ln (x y)}}{y} + e^{x \ln (x y)} \ln (x y) + e^{x \ln (x y)} = 0$

Step 5.2

Reorder factors in $\frac{y' x e^{x \ln (x y)}}{y} + e^{x \ln (x y)} \ln (x y) + e^{x \ln (x y)}$ .

$\frac{y' x e^{x \ln (x y)}}{y} + \ln (x y) e^{x \ln (x y)} + e^{x \ln (x y)} = 0$

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.1

Subtract $\ln (x y) e^{x \ln (x y)}$ from both sides of the equation.

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

Step 5.3.2

Subtract $e^{x \ln (x y)}$ from both sides of the equation.

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

Step 5.4

Multiply both sides by $y$ .

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

Step 5.5

Simplify.

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Step 5.5.1

Simplify the left side.

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Step 5.5.1.1

Cancel the common factor of $y$ .

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Step 5.5.1.1.1

Cancel the common factor.

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

Step 5.5.1.1.2

Rewrite the expression.

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

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Simplify $(- \ln (x y) e^{x \ln (x y)} - e^{x \ln (x y)}) y$ .

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Step 5.5.2.1.1

Apply the distributive property.

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

Step 5.5.2.1.2

Simplify the expression.

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Step 5.5.2.1.2.1

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

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

Step 5.5.2.1.2.2

Move $\ln (x y)$ .

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

Step 5.6

Divide each term in $y' x e^{x \ln (x y)} = - y \ln (x y) e^{x \ln (x y)} - y e^{x \ln (x y)}$ by $x e^{x \ln (x y)}$ and simplify.

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Step 5.6.1

Divide each term in $y' x e^{x \ln (x y)} = - y \ln (x y) e^{x \ln (x y)} - y e^{x \ln (x y)}$ by $x e^{x \ln (x y)}$ .

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $x$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Rewrite the expression.

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

Step 5.6.2.2

Cancel the common factor of $e^{x \ln (x y)}$ .

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Step 5.6.2.2.1

Cancel the common factor.

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

Step 5.6.2.2.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Simplify each term.

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Step 5.6.3.1.1

Cancel the common factor of $e^{x \ln (x y)}$ .

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Step 5.6.3.1.1.1

Cancel the common factor.

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

Step 5.6.3.1.1.2

Rewrite the expression.

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

Step 5.6.3.1.2

Move the negative in front of the fraction.

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

Step 5.6.3.1.3

Cancel the common factor of $e^{x \ln (x y)}$ .

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Step 5.6.3.1.3.1

Cancel the common factor.

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

Step 5.6.3.1.3.2

Rewrite the expression.

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

Step 5.6.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

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