Algebra Examples

$z = x^{y} + \ln (x y)$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x^{y} + \ln (x y)$ with respect to $y$ is $\frac{d}{d y} [x^{y}] + \frac{d}{d y} [\ln (x y)]$ .

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

Step 3.2

Evaluate $\frac{d}{d y} [x^{y}]$ .

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Step 3.2.1

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 $f = x$ and $g = y$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.3

Evaluate $\frac{d}{d y} [\ln (x y)]$ .

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Step 3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \ln (y)$ and $g (y) = x y$ .

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Step 3.3.1.1

To apply the Chain Rule, set $u$ as $x y$ .

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

Step 3.3.1.2

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

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

Step 3.3.1.3

Replace all occurrences of $u$ with $x y$ .

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

Step 3.3.2

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 $g (y) = y$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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

Step 3.3.5

Multiply $x$ by $1$ .

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

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

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

Step 3.4.2

Combine terms.

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Step 3.4.2.1

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

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

Step 3.4.2.2

Cancel the common factor of $x$ .

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Step 3.4.2.2.1

Cancel the common factor.

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

Step 3.4.2.2.2

Rewrite the expression.

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

Step 3.4.2.3

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

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

Step 3.4.2.4

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

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

Step 3.4.2.5

Cancel the common factor of $y$ .

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Step 3.4.2.5.1

Cancel the common factor.

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

Step 3.4.2.5.2

Rewrite the expression.

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

Step 3.4.2.6

To write $x' (y x^{y - 1})$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3.4.2.7

Combine the numerators over the common denominator.

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

Step 3.4.2.8

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

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

Step 3.4.2.9

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

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

Step 3.4.2.10

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

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

Step 3.4.2.11

Add $1$ and $1$ .

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

Step 3.4.2.12

To write $x^{y} \ln (x)$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3.4.2.13

Combine the numerators over the common denominator.

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

Step 3.4.2.14

To write $\frac{x^{y} \ln (x) y + x' (x^{y - 1}) y^{2} + 1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.4.2.15

To write $\frac{x'}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3.4.2.16

Write each expression with a common denominator of $y x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.2.16.1

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

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

Step 3.4.2.16.2

Multiply $\frac{x'}{x}$ by $\frac{y}{y}$ .

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

Step 3.4.2.16.3

Reorder the factors of $y x$ .

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

Step 3.4.2.17

Combine the numerators over the common denominator.

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

Step 3.4.3

Reorder terms.

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

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Apply the distributive property.

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

Step 3.4.4.2

Simplify.

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Step 3.4.4.2.1

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

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Step 3.4.4.2.1.1

Move $x^{y}$ .

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

Step 3.4.4.2.1.2

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

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Step 3.4.4.2.1.2.1

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

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

Step 3.4.4.2.1.2.2

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

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

Step 3.4.4.2.2

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

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Step 3.4.4.2.2.1

Move $x^{y - 1}$ .

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

Step 3.4.4.2.2.2

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

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Step 3.4.4.2.2.2.1

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

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

Step 3.4.4.2.2.2.2

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

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

Step 3.4.4.2.2.3

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

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Step 3.4.4.2.2.3.1

Add $- 1$ and $1$ .

$\frac{x^{y + 1} (\ln (x) y) + x^{y + 0} (x' y^{2}) + x \cdot 1 + y x'}{x y}$

Step 3.4.4.2.2.3.2

Add $y$ and $0$ .

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

Step 3.4.4.2.3

Multiply $x$ by $1$ .

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

Step 3.4.5

Reorder factors in $\frac{x^{y + 1} (\ln (x) y) + x^{y} (x' y^{2}) + x + y x'}{x y}$ .

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

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Set the numerator equal to zero.

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

Step 5.2

Solve the equation for $x'$ .

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Step 5.2.1

Reorder factors in $y x^{y + 1} \ln (x) + y^{2} x^{y} x' + x + y x'$ .

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

Step 5.2.2

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

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Step 5.2.2.1

Subtract $y x^{y + 1} \ln (x)$ from both sides of the equation.

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

Step 5.2.2.2

Subtract $x$ from both sides of the equation.

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

Step 5.2.3

Factor $y x'$ out of $y^{2} x' x^{y} + y x'$ .

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Step 5.2.3.1

Factor $y x'$ out of $y^{2} x' x^{y}$ .

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

Step 5.2.3.2

Factor $y x'$ out of $y x'$ .

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

Step 5.2.3.3

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

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

Step 5.2.4

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

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Step 5.2.4.1

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

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

Step 5.2.4.2

Simplify the left side.

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Step 5.2.4.2.1

Cancel the common factor of $y$ .

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Step 5.2.4.2.1.1

Cancel the common factor.

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

Step 5.2.4.2.1.2

Rewrite the expression.

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

Step 5.2.4.2.2

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

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Step 5.2.4.2.2.1

Cancel the common factor.

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

Step 5.2.4.2.2.2

Divide $x'$ by $1$ .

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

Step 5.2.4.3

Simplify the right side.

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Step 5.2.4.3.1

Combine the numerators over the common denominator.

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

Step 5.2.4.3.2

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

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

Step 5.2.4.3.3

Factor $- 1$ out of $- x$ .

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

Step 5.2.4.3.4

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

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

Step 5.2.4.3.5

Simplify the expression.

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Step 5.2.4.3.5.1

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

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

Step 5.2.4.3.5.2

Move the negative in front of the fraction.

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

Step 6

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

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