Calculus Examples

$x^{y} = y^{x}$

Step 1

Differentiate both sides of the equation.

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

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 $\frac{d}{d x} [f^{g}]$ is $f^{g} (f' \frac{g}{f} + g' \ln (f))$ where $f = x$ and $g = y$ .

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

Step 2.2

Differentiate using the Power Rule.

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Step 2.2.1

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

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

Step 2.2.2

Multiply $y$ by $1$ .

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

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

Step 2.4.2

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify.

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Step 3.5.1

Reorder terms.

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

Step 3.5.2

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Move all the terms containing a logarithm to the left side of the equation.

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

Step 5.2

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

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

Step 5.3

Reorder factors in $- y x^{y - 1} + x y^{x - 1} y'$ .

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Add $\ln (y) y^{x}$ to both sides of the equation.

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

Step 5.7

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

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Step 5.7.1

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

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

Step 5.7.2

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

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

Step 5.7.3

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

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

Step 5.8

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

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Step 5.8.1

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

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

Step 5.8.2

Simplify the left side.

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Step 5.8.2.1

Cancel the common factor of $\ln (x) x^{y} - x y^{x - 1}$ .

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Step 5.8.2.1.1

Cancel the common factor.

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

Step 5.8.2.1.2

Divide $y'$ by $1$ .

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

Step 5.8.3

Simplify the right side.

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Step 5.8.3.1

Combine the numerators over the common denominator.

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

Step 5.8.3.2

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

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

Step 5.8.3.3

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

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

Step 5.8.3.4

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

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

Step 5.8.3.5

Simplify the expression.

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Step 5.8.3.5.1

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

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

Step 5.8.3.5.2

Move the negative in front of the fraction.

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

Step 6

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

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