Calculus Examples

$x^{5 y} = y^{6 x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{5 y}) = \frac{d}{d x} (y^{6 x})$

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 $g = 5 y$ .

$\frac{d}{d x} [x] (5 y x^{5 y - 1}) + \frac{d}{d x} [5 y] (x^{5 y} \ln (x))$

Step 2.2

Differentiate.

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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 $n = 1$ .

$1 (5 y x^{5 y - 1}) + \frac{d}{d x} [5 y] (x^{5 y} \ln (x))$

Step 2.2.2

Multiply $5$ by $1$ .

$5 (y x^{5 y - 1}) + \frac{d}{d x} [5 y] (x^{5 y} \ln (x))$

Step 2.2.3

Since $5$ is constant with respect to $x$ , the derivative of $5 y$ with respect to $x$ is $5 \frac{d}{d x} [y]$ .

$5 y x^{5 y - 1} + 5 \frac{d}{d x} [y] (x^{5 y} \ln (x))$

Step 2.3

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

$5 y x^{5 y - 1} + 5 y' (x^{5 y} \ln (x))$

Step 2.4

Simplify.

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Step 2.4.1

Reorder terms.

$5 x^{5 y - 1} y + 5 x^{5 y} \ln (x) y'$

Step 2.4.2

Reorder factors in $5 x^{5 y - 1} y + 5 x^{5 y} \ln (x) y'$ .

$5 y x^{5 y - 1} + 5 x^{5 y} \ln (x) y'$

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 $g = 6 x$ .

$\frac{d}{d x} [y] (6 x y^{6 x - 1}) + \frac{d}{d x} [6 x] (y^{6 x} \ln (y))$

Step 3.2

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

$y' (6 x y^{6 x - 1}) + \frac{d}{d x} [6 x] (y^{6 x} \ln (y))$

Step 3.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$y' (6 x y^{6 x - 1}) + 6 \frac{d}{d x} [x] (y^{6 x} \ln (y))$

Step 3.4

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

$y' (6 x y^{6 x - 1}) + 6 \cdot 1 (y^{6 x} \ln (y))$

Step 3.5

Multiply $6$ by $1$ .

$y' (6 x y^{6 x - 1}) + 6 (y^{6 x} \ln (y))$

Step 3.6

Simplify.

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Step 3.6.1

Reorder terms.

$6 y^{6 x - 1} x y' + 6 y^{6 x} \ln (y)$

Step 3.6.2

Reorder factors in $6 y^{6 x - 1} x y' + 6 y^{6 x} \ln (y)$ .

$6 x y^{6 x - 1} y' + 6 y^{6 x} \ln (y)$

Step 4

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

$5 y x^{5 y - 1} + 5 x^{5 y} \ln (x) y' = 6 x y^{6 x - 1} y' + 6 y^{6 x} \ln (y)$

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.

$5 x^{5 y} \ln (x) y' - 6 y^{6 x} \ln (y) = - 5 y x^{5 y - 1} + 6 x y^{6 x - 1} y'$

Step 5.2

Reorder factors in $5 x^{5 y} \ln (x) y' - 6 y^{6 x} \ln (y)$ .

$5 y' x^{5 y} \ln (x) - 6 y^{6 x} \ln (y) = - 5 y x^{5 y - 1} + 6 x y^{6 x - 1} y'$

Step 5.3

Reorder factors in $- 5 y x^{5 y - 1} + 6 x y^{6 x - 1} y'$ .

$5 y' x^{5 y} \ln (x) - 6 y^{6 x} \ln (y) = - 5 y x^{5 y - 1} + 6 x y' y^{6 x - 1}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Simplify $5 \ln (x)$ by moving $5$ inside the logarithm.

$y' x^{5 y} \ln (x^{5}) - 6 y^{6 x} \ln (y) = - 5 y x^{5 y - 1} + 6 x y' y^{6 x - 1}$

Step 5.4.1.2

Simplify $- 6 \ln (y)$ by moving $6$ inside the logarithm.

$y' x^{5 y} \ln (x^{5}) + y^{6 x} (- \ln (y^{6})) = - 5 y x^{5 y - 1} + 6 x y' y^{6 x - 1}$

Step 5.4.1.3

Rewrite using the commutative property of multiplication.

$y' x^{5 y} \ln (x^{5}) - y^{6 x} \ln (y^{6}) = - 5 y x^{5 y - 1} + 6 x y' y^{6 x - 1}$

Step 5.5

Reorder factors in $y' x^{5 y} \ln (x^{5}) - y^{6 x} \ln (y^{6})$ .

$y' \ln (x^{5}) x^{5 y} - \ln (y^{6}) y^{6 x} = - 5 y x^{5 y - 1} + 6 x y' y^{6 x - 1}$

Step 5.6

Subtract $6 x y' y^{6 x - 1}$ from both sides of the equation.

$y' \ln (x^{5}) x^{5 y} - \ln (y^{6}) y^{6 x} - 6 x y' y^{6 x - 1} = - 5 y x^{5 y - 1}$

Step 5.7

Add $\ln (y^{6}) y^{6 x}$ to both sides of the equation.

$y' \ln (x^{5}) x^{5 y} - 6 x y' y^{6 x - 1} = - 5 y x^{5 y - 1} + \ln (y^{6}) y^{6 x}$

Step 5.8

Factor $y'$ out of $y' \ln (x^{5}) x^{5 y} - 6 x y' y^{6 x - 1}$ .

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Step 5.8.1

Factor $y'$ out of $y' \ln (x^{5}) x^{5 y}$ .

$y' (\ln (x^{5}) x^{5 y}) - 6 x y' y^{6 x - 1} = - 5 y x^{5 y - 1} + \ln (y^{6}) y^{6 x}$

Step 5.8.2

Factor $y'$ out of $- 6 x y' y^{6 x - 1}$ .

$y' (\ln (x^{5}) x^{5 y}) + y' (- 6 x y^{6 x - 1}) = - 5 y x^{5 y - 1} + \ln (y^{6}) y^{6 x}$

Step 5.8.3

Factor $y'$ out of $y' (\ln (x^{5}) x^{5 y}) + y' (- 6 x y^{6 x - 1})$ .

$y' (\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}) = - 5 y x^{5 y - 1} + \ln (y^{6}) y^{6 x}$

Step 5.9

Divide each term in $y' (\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}) = - 5 y x^{5 y - 1} + \ln (y^{6}) y^{6 x}$ by $\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}$ and simplify.

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Step 5.9.1

Divide each term in $y' (\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}) = - 5 y x^{5 y - 1} + \ln (y^{6}) y^{6 x}$ by $\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}$ .

$\frac{y' (\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1})}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}} = \frac{- 5 y x^{5 y - 1}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}} + \frac{\ln (y^{6}) y^{6 x}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 5.9.2

Simplify the left side.

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Step 5.9.2.1

Cancel the common factor of $\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}$ .

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Step 5.9.2.1.1

Cancel the common factor.

Step 5.9.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 5 y x^{5 y - 1}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}} + \frac{\ln (y^{6}) y^{6 x}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 5.9.3

Simplify the right side.

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Step 5.9.3.1

Combine the numerators over the common denominator.

$y' = \frac{- 5 y x^{5 y - 1} + \ln (y^{6}) y^{6 x}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 5.9.3.2

Factor $- 1$ out of $- 5 y x^{5 y - 1}$ .

$y' = \frac{- (5 y x^{5 y - 1}) + \ln (y^{6}) y^{6 x}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 5.9.3.3

Factor $- 1$ out of $\ln (y^{6}) y^{6 x}$ .

$y' = \frac{- (5 y x^{5 y - 1}) - (- \ln (y^{6}) y^{6 x})}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 5.9.3.4

Factor $- 1$ out of $- (5 y x^{5 y - 1}) - (- \ln (y^{6}) y^{6 x})$ .

$y' = \frac{- (5 y x^{5 y - 1} - \ln (y^{6}) y^{6 x})}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 5.9.3.5

Simplify the expression.

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Step 5.9.3.5.1

Rewrite $- (5 y x^{5 y - 1} - \ln (y^{6}) y^{6 x})$ as $- 1 (5 y x^{5 y - 1} - \ln (y^{6}) y^{6 x})$ .

$y' = \frac{- 1 (5 y x^{5 y - 1} - \ln (y^{6}) y^{6 x})}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 5.9.3.5.2

Move the negative in front of the fraction.

$y' = - \frac{5 y x^{5 y - 1} - \ln (y^{6}) y^{6 x}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$

Step 6

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

$\frac{d y}{d x} = - \frac{5 y x^{5 y - 1} - \ln (y^{6}) y^{6 x}}{\ln (x^{5}) x^{5 y} - 6 x y^{6 x - 1}}$