Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Generalized Power Rule which states that $\frac{d}{d y} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = x y$ and $g = x$ .

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

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

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

Step 2.3.2

Multiply $x$ by $1$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the product rule to $x y$ .

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

Step 2.6.2

Apply the product rule to $x y$ .

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

Step 2.6.3

Apply the distributive property.

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

Step 2.6.4

Combine terms.

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Step 2.6.4.1

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

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

Step 2.6.4.2

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

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

Step 2.6.4.3

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

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

Step 2.6.4.4

Add $1$ and $1$ .

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

Step 2.6.5

Reorder terms.

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

Step 3

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

$0$

Step 4

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

$x^{x - 1} y^{x - 1} (x^{2} + y x' x) + x^{x} y^{x} \ln (x y) x' = 0$

Step 5

Solve for $x'$ .

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Step 5.1

Simplify $x^{x - 1} y^{x - 1} (x^{2} + y x' x) + x^{x} y^{x} \ln (x y) x'$ .

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the distributive property.

$x^{x - 1} y^{x - 1} x^{2} + x^{x - 1} y^{x - 1} (y x' x) + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.2

Multiply $x^{x - 1}$ by $x^{2}$ by adding the exponents.

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Step 5.1.1.2.1

Move $x^{2}$ .

$x^{2} x^{x - 1} y^{x - 1} + x^{x - 1} y^{x - 1} (y x' x) + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.2.2

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

$x^{2 + x - 1} y^{x - 1} + x^{x - 1} y^{x - 1} (y x' x) + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.2.3

Subtract $1$ from $2$ .

$x^{x + 1} y^{x - 1} + x^{x - 1} y^{x - 1} (y x' x) + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.3

Multiply $x^{x - 1}$ by $x$ by adding the exponents.

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Step 5.1.1.3.1

Move $x$ .

$x^{x + 1} y^{x - 1} + x \cdot x^{x - 1} y^{x - 1} (y x') + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.3.2

Multiply $x$ by $x^{x - 1}$ .

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Step 5.1.1.3.2.1

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

$x^{x + 1} y^{x - 1} + x^{1} x^{x - 1} y^{x - 1} (y x') + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.3.2.2

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

$x^{x + 1} y^{x - 1} + x^{1 + x - 1} y^{x - 1} (y x') + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.3.3

Combine the opposite terms in $1 + x - 1$ .

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Step 5.1.1.3.3.1

Subtract $1$ from $1$ .

$x^{x + 1} y^{x - 1} + x^{x + 0} y^{x - 1} (y x') + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.3.3.2

Add $x$ and $0$ .

$x^{x + 1} y^{x - 1} + x^{x} y^{x - 1} (y x') + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.4

Multiply $y^{x - 1}$ by $y$ by adding the exponents.

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Step 5.1.1.4.1

Move $y$ .

$x^{x + 1} y^{x - 1} + x^{x} (y \cdot y^{x - 1}) x' + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.4.2

Multiply $y$ by $y^{x - 1}$ .

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Step 5.1.1.4.2.1

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

$x^{x + 1} y^{x - 1} + x^{x} (y^{1} y^{x - 1}) x' + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.4.2.2

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

$x^{x + 1} y^{x - 1} + x^{x} y^{1 + x - 1} x' + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.4.3

Combine the opposite terms in $1 + x - 1$ .

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Step 5.1.1.4.3.1

Subtract $1$ from $1$ .

$x^{x + 1} y^{x - 1} + x^{x} y^{x + 0} x' + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.1.4.3.2

Add $x$ and $0$ .

$x^{x + 1} y^{x - 1} + x^{x} y^{x} x' + x^{x} y^{x} \ln (x y) x' = 0$

Step 5.1.2

Reorder factors in $x^{x + 1} y^{x - 1} + x^{x} y^{x} x' + x^{x} y^{x} \ln (x y) x'$ .

$x^{x + 1} y^{x - 1} + x' x^{x} y^{x} + x' x^{x} y^{x} \ln (x y) = 0$

Step 5.2

Subtract $x^{x + 1} y^{x - 1}$ from both sides of the equation.

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

Step 5.3

Factor $x' x^{x} y^{x}$ out of $x' x^{x} y^{x} + x' x^{x} y^{x} \ln (x y)$ .

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Step 5.3.1

Factor $x' x^{x} y^{x}$ out of $x' x^{x} y^{x}$ .

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

Step 5.3.2

Factor $x' x^{x} y^{x}$ out of $x' x^{x} y^{x} \ln (x y)$ .

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

Step 5.3.3

Factor $x' x^{x} y^{x}$ out of $x' x^{x} y^{x} \cdot 1 + x' x^{x} y^{x} \ln (x y)$ .

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

Step 5.4

Divide each term in $x' x^{x} y^{x} (1 + \ln (x y)) = - x^{x + 1} y^{x - 1}$ by $x^{x} y^{x} (1 + \ln (x y))$ and simplify.

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Step 5.4.1

Divide each term in $x' x^{x} y^{x} (1 + \ln (x y)) = - x^{x + 1} y^{x - 1}$ by $x^{x} y^{x} (1 + \ln (x y))$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x^{x}$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Rewrite the expression.

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

Step 5.4.2.2

Cancel the common factor of $y^{x}$ .

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Step 5.4.2.2.1

Cancel the common factor.

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

Step 5.4.2.2.2

Rewrite the expression.

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

Step 5.4.2.3

Cancel the common factor of $1 + \ln (x y)$ .

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Step 5.4.2.3.1

Cancel the common factor.

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

Step 5.4.2.3.2

Divide $x'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $x^{x + 1}$ and $x^{x}$ .

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Step 5.4.3.1.1

Factor $x^{x}$ out of $- x^{x + 1} y^{x - 1}$ .

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

Step 5.4.3.1.2

Cancel the common factors.

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Step 5.4.3.1.2.1

Factor $x^{x}$ out of $x^{x} y^{x} (1 + \ln (x y))$ .

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

Step 5.4.3.1.2.2

Cancel the common factor.

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

Step 5.4.3.1.2.3

Rewrite the expression.

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

Step 5.4.3.2

Cancel the common factor of $y^{x - 1}$ and $y^{x}$ .

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Step 5.4.3.2.1

Factor $y^{x}$ out of $- x y^{x - 1}$ .

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

Step 5.4.3.2.2

Cancel the common factors.

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Step 5.4.3.2.2.1

Cancel the common factor.

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

Step 5.4.3.2.2.2

Rewrite the expression.

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

Step 5.4.3.3

Simplify the expression.

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Step 5.4.3.3.1

Move $y^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.4.3.3.2

Move the negative in front of the fraction.

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

Step 5.4.3.3.3

Reorder factors in $- \frac{x}{(1 + \ln (x y)) y}$ .

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

Step 6

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

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