Algebra Examples

$\frac{x^{2}}{x + y} = y^{2} + 6$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (\frac{x^{2}}{x + y}) = \frac{d}{d y} (y^{2} + 6)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = x^{2}$ and $g (y) = x + y$ .

$\frac{(x + y) \frac{d}{d y} [x^{2}] - x^{2} \frac{d}{d y} [x + y]}{{(x + y)}^{2}}$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 2.2.1

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

$\frac{(x + y) (\frac{d}{d u} [u^{2}] \frac{d}{d y} [x]) - x^{2} \frac{d}{d y} [x + y]}{{(x + y)}^{2}}$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$\frac{(x + y) (2 u \frac{d}{d y} [x]) - x^{2} \frac{d}{d y} [x + y]}{{(x + y)}^{2}}$

Step 2.2.3

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

$\frac{(x + y) (2 x \frac{d}{d y} [x]) - x^{2} \frac{d}{d y} [x + y]}{{(x + y)}^{2}}$

Step 2.3

Move $2$ to the left of $x + y$ .

$\frac{2 \cdot (x + y) x \frac{d}{d y} [x] - x^{2} \frac{d}{d y} [x + y]}{{(x + y)}^{2}}$

Step 2.4

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

$\frac{2 (x + y) x x' - x^{2} \frac{d}{d y} [x + y]}{{(x + y)}^{2}}$

Step 2.5

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

$\frac{2 (x + y) x x' - x^{2} (\frac{d}{d y} [x] + \frac{d}{d y} [y])}{{(x + y)}^{2}}$

Step 2.6

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

$\frac{2 (x + y) x x' - x^{2} (x' + \frac{d}{d y} [y])}{{(x + y)}^{2}}$

Step 2.7

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

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Apply the distributive property.

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

Step 2.8.4

Apply the distributive property.

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

Step 2.8.5

Simplify the numerator.

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Step 2.8.5.1

Simplify each term.

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Step 2.8.5.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.8.5.1.1.1

Move $x$ .

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

Step 2.8.5.1.1.2

Multiply $x$ by $x$ .

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

Step 2.8.5.1.2

Multiply $- 1$ by $1$ .

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

Step 2.8.5.2

Subtract $x^{2} x'$ from $2 x^{2} x'$ .

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

Step 2.8.5.3

Multiply $x^{2}$ by $1$ .

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

Step 2.8.6

Reorder terms.

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

Step 2.8.7

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

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Step 2.8.7.1

Factor $x$ out of $- x^{2}$ .

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

Step 2.8.7.2

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

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

Step 2.8.7.3

Factor $x$ out of $2 x y x'$ .

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

Step 2.8.7.4

Factor $x$ out of $x (- x) + x (x x')$ .

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

Step 2.8.7.5

Factor $x$ out of $x (- x + x x') + x (2 y x')$ .

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

Step 2.8.8

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

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

Step 2.8.9

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

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

Step 2.8.10

Factor $- 1$ out of $- (x) - (- x x')$ .

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

Step 2.8.11

Factor $- 1$ out of $2 y x'$ .

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

Step 2.8.12

Factor $- 1$ out of $- (x - x x') - (- 2 y x')$ .

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

Step 2.8.13

Rewrite $- (x - x x' - 2 y x')$ as $- 1 (x - x x' - 2 y x')$ .

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

Step 2.8.14

Move the negative in front of the fraction.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $y^{2} + 6$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [6]$ .

$\frac{d}{d y} [y^{2}] + \frac{d}{d y} [6]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$2 y + \frac{d}{d y} [6]$

Step 3.3

Since $6$ is constant with respect to $y$ , the derivative of $6$ with respect to $y$ is $0$ .

$2 y + 0$

Step 3.4

Add $2 y$ and $0$ .

$2 y$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Divide each term in $- \frac{x (x - x x' - 2 y x')}{{(x + y)}^{2}} = 2 y$ by $- 1$ and simplify.

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Step 5.1.1

Divide each term in $- \frac{x (x - x x' - 2 y x')}{{(x + y)}^{2}} = 2 y$ by $- 1$ .

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

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Dividing two negative values results in a positive value.

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

Step 5.1.2.2

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

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

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Move the negative one from the denominator of $\frac{2 y}{- 1}$ .

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

Step 5.1.3.2

Rewrite $- 1 \cdot (2 y)$ as $- (2 y)$ .

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

Step 5.1.3.3

Multiply $2$ by $- 1$ .

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

Step 5.2

Multiply both sides by ${(x + y)}^{2}$ .

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

Step 5.3

Simplify the left side.

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Step 5.3.1

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

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Step 5.3.1.1

Cancel the common factor of ${(x + y)}^{2}$ .

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Step 5.3.1.1.1

Cancel the common factor.

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

Step 5.3.1.1.2

Rewrite the expression.

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

Step 5.3.1.2

Apply the distributive property.

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

Step 5.3.1.3

Simplify.

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Step 5.3.1.3.1

Multiply $x$ by $x$ .

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

Step 5.3.1.3.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.3.3

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.4.1

Move $x$ .

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

Step 5.3.1.4.2

Multiply $x$ by $x$ .

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

Step 5.3.1.5

Move $x^{2}$ .

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

Step 5.3.1.6

Move $y$ .

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

Step 5.3.1.7

Move $x$ .

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

Step 5.3.1.8

Move $x^{2}$ .

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

Step 5.3.1.9

Reorder $- x' x^{2}$ and $- 2 x' x y$ .

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

Step 5.4

Solve for $x'$ .

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Step 5.4.1

Simplify $- 2 y {(x + y)}^{2}$ .

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Step 5.4.1.1

Rewrite.

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

Step 5.4.1.2

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

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

Step 5.4.1.3

Expand $(x + y) (x + y)$ using the FOIL Method.

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Step 5.4.1.3.1

Apply the distributive property.

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

Step 5.4.1.3.2

Apply the distributive property.

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

Step 5.4.1.3.3

Apply the distributive property.

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

Step 5.4.1.4

Simplify and combine like terms.

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Step 5.4.1.4.1

Simplify each term.

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Step 5.4.1.4.1.1

Multiply $x$ by $x$ .

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

Step 5.4.1.4.1.2

Multiply $y$ by $y$ .

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

Step 5.4.1.4.2

Add $x y$ and $y x$ .

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Step 5.4.1.4.2.1

Reorder $y$ and $x$ .

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

Step 5.4.1.4.2.2

Add $x y$ and $x y$ .

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

Step 5.4.1.5

Apply the distributive property.

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

Step 5.4.1.6

Simplify.

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Step 5.4.1.6.1

Multiply $y$ by $y$ by adding the exponents.

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Step 5.4.1.6.1.1

Move $y$ .

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

Step 5.4.1.6.1.2

Multiply $y$ by $y$ .

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

Step 5.4.1.6.2

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 5.4.1.6.2.1

Move $y^{2}$ .

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

Step 5.4.1.6.2.2

Multiply $y^{2}$ by $y$ .

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Step 5.4.1.6.2.2.1

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

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

Step 5.4.1.6.2.2.2

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

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

Step 5.4.1.6.2.3

Add $2$ and $1$ .

$- 2 x' x y - x' x^{2} + x^{2} = - 2 y x^{2} - 2 y^{2} (2 x) - 2 y^{3}$

Step 5.4.1.7

Simplify each term.

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Step 5.4.1.7.1

Rewrite using the commutative property of multiplication.

$- 2 x' x y - x' x^{2} + x^{2} = - 2 y x^{2} - 2 \cdot 2 y^{2} x - 2 y^{3}$

Step 5.4.1.7.2

Multiply $- 2$ by $2$ .

$- 2 x' x y - x' x^{2} + x^{2} = - 2 y x^{2} - 4 y^{2} x - 2 y^{3}$

Step 5.4.2

Subtract $x^{2}$ from both sides of the equation.

$- 2 x' x y - x' x^{2} = - 2 y x^{2} - 4 y^{2} x - 2 y^{3} - x^{2}$

Step 5.4.3

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

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Step 5.4.3.1

Factor $x' x$ out of $- 2 x' x y$ .

$x' x (- 2 y) - x' x^{2} = - 2 y x^{2} - 4 y^{2} x - 2 y^{3} - x^{2}$

Step 5.4.3.2

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

$x' x (- 2 y) + x' x (- 1 x) = - 2 y x^{2} - 4 y^{2} x - 2 y^{3} - x^{2}$

Step 5.4.3.3

Factor $x' x$ out of $x' x (- 2 y) + x' x (- 1 x)$ .

$x' x (- 2 y - 1 x) = - 2 y x^{2} - 4 y^{2} x - 2 y^{3} - x^{2}$

Step 5.4.4

Rewrite $- 1 x$ as $- x$ .

$x' x (- 2 y - x) = - 2 y x^{2} - 4 y^{2} x - 2 y^{3} - x^{2}$

Step 5.4.5

Divide each term in $x' x (- 2 y - x) = - 2 y x^{2} - 4 y^{2} x - 2 y^{3} - x^{2}$ by $x (- 2 y - x)$ and simplify.

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Step 5.4.5.1

Divide each term in $x' x (- 2 y - x) = - 2 y x^{2} - 4 y^{2} x - 2 y^{3} - x^{2}$ by $x (- 2 y - x)$ .

$\frac{x' x (- 2 y - x)}{x (- 2 y - x)} = \frac{- 2 y x^{2}}{x (- 2 y - x)} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.2

Simplify the left side.

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Step 5.4.5.2.1

Cancel the common factor of $x$ .

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Step 5.4.5.2.1.1

Cancel the common factor.

$\frac{x' x (- 2 y - x)}{x (- 2 y - x)} = \frac{- 2 y x^{2}}{x (- 2 y - x)} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.2.1.2

Rewrite the expression.

$\frac{x' (- 2 y - x)}{- 2 y - x} = \frac{- 2 y x^{2}}{x (- 2 y - x)} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.2.2

Cancel the common factor of $- 2 y - x$ .

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Step 5.4.5.2.2.1

Cancel the common factor.

$\frac{x' (- 2 y - x)}{- 2 y - x} = \frac{- 2 y x^{2}}{x (- 2 y - x)} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{- 2 y x^{2}}{x (- 2 y - x)} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3

Simplify the right side.

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Step 5.4.5.3.1

Simplify each term.

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Step 5.4.5.3.1.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.4.5.3.1.1.1

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

$x' = \frac{x (- 2 y x)}{x (- 2 y - x)} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.1.2

Cancel the common factors.

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Step 5.4.5.3.1.1.2.1

Cancel the common factor.

$x' = \frac{x (- 2 y x)}{x (- 2 y - x)} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.1.2.2

Rewrite the expression.

$x' = \frac{- 2 y x}{- 2 y - x} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.2

Move the negative in front of the fraction.

$x' = - \frac{2 y x}{- 2 y - x} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.3

Cancel the common factor of $x$ .

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Step 5.4.5.3.1.3.1

Cancel the common factor.

$x' = - \frac{2 y x}{- 2 y - x} + \frac{- 4 y^{2} x}{x (- 2 y - x)} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.3.2

Rewrite the expression.

$x' = - \frac{2 y x}{- 2 y - x} + \frac{- 4 y^{2}}{- 2 y - x} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.4

Move the negative in front of the fraction.

$x' = - \frac{2 y x}{- 2 y - x} - \frac{4 y^{2}}{- 2 y - x} + \frac{- 2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.5

Move the negative in front of the fraction.

$x' = - \frac{2 y x}{- 2 y - x} - \frac{4 y^{2}}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{- x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.1.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.4.5.3.1.6.1

Factor $x$ out of $- x^{2}$ .

$x' = - \frac{2 y x}{- 2 y - x} - \frac{4 y^{2}}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{x (- x)}{x (- 2 y - x)}$

Step 5.4.5.3.1.6.2

Cancel the common factors.

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Step 5.4.5.3.1.6.2.1

Cancel the common factor.

$x' = - \frac{2 y x}{- 2 y - x} - \frac{4 y^{2}}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{x (- x)}{x (- 2 y - x)}$

Step 5.4.5.3.1.6.2.2

Rewrite the expression.

$x' = - \frac{2 y x}{- 2 y - x} - \frac{4 y^{2}}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{- x}{- 2 y - x}$

Step 5.4.5.3.1.7

Move the negative in front of the fraction.

$x' = - \frac{2 y x}{- 2 y - x} - \frac{4 y^{2}}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.2

Combine the numerators over the common denominator.

$x' = \frac{- 2 y x - 4 y^{2}}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.3

Simplify each term.

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Step 5.4.5.3.3.1

Simplify the numerator.

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Step 5.4.5.3.3.1.1

Factor $2 y$ out of $- 2 y x - 4 y^{2}$ .

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Step 5.4.5.3.3.1.1.1

Factor $2 y$ out of $- 2 y x$ .

$x' = \frac{2 y (- 1 x) - 4 y^{2}}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.3.1.1.2

Factor $2 y$ out of $- 4 y^{2}$ .

$x' = \frac{2 y (- 1 x) + 2 y (- 2 y)}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.3.1.1.3

Factor $2 y$ out of $2 y (- 1 x) + 2 y (- 2 y)$ .

$x' = \frac{2 y (- 1 x - 2 y)}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.3.1.2

Rewrite $- 1 x$ as $- x$ .

$x' = \frac{2 y (- x - 2 y)}{- 2 y - x} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.3.2

Cancel the common factor of $- x - 2 y$ and $- 2 y - x$ .

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Step 5.4.5.3.3.2.1

Reorder terms.

$x' = \frac{2 y (- x - 2 y)}{- x - 2 y} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.3.2.2

Cancel the common factor.

$x' = \frac{2 y (- x - 2 y)}{- x - 2 y} - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.3.2.3

Divide $2 y$ by $1$ .

$x' = 2 y - \frac{2 y^{3}}{x (- 2 y - x)} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.4

To write $2 y$ as a fraction with a common denominator, multiply by $\frac{- 2 y - x}{- 2 y - x}$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{2 y (- 2 y - x)}{- 2 y - x} - \frac{x}{- 2 y - x}$

Step 5.4.5.3.5

Combine the numerators over the common denominator.

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{2 y (- 2 y - x) - x}{- 2 y - x}$

Step 5.4.5.3.6

Simplify the numerator.

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Step 5.4.5.3.6.1

Apply the distributive property.

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{2 y (- 2 y) + 2 y (- x) - x}{- 2 y - x}$

Step 5.4.5.3.6.2

Rewrite using the commutative property of multiplication.

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{2 \cdot - 2 y \cdot y + 2 y (- x) - x}{- 2 y - x}$

Step 5.4.5.3.6.3

Rewrite using the commutative property of multiplication.

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{2 \cdot - 2 y \cdot y + 2 \cdot - 1 y x - x}{- 2 y - x}$

Step 5.4.5.3.6.4

Simplify each term.

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Step 5.4.5.3.6.4.1

Multiply $y$ by $y$ by adding the exponents.

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Step 5.4.5.3.6.4.1.1

Move $y$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{2 \cdot - 2 (y \cdot y) + 2 \cdot - 1 y x - x}{- 2 y - x}$

Step 5.4.5.3.6.4.1.2

Multiply $y$ by $y$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{2 \cdot - 2 y^{2} + 2 \cdot - 1 y x - x}{- 2 y - x}$

Step 5.4.5.3.6.4.2

Multiply $2$ by $- 2$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{- 4 y^{2} + 2 \cdot - 1 y x - x}{- 2 y - x}$

Step 5.4.5.3.6.4.3

Multiply $2$ by $- 1$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{- 4 y^{2} - 2 y x - x}{- 2 y - x}$

Step 5.4.5.3.7

To write $\frac{- 4 y^{2} - 2 y x - x}{- 2 y - x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{- 4 y^{2} - 2 y x - x}{- 2 y - x} \cdot \frac{x}{x}$

Step 5.4.5.3.8

Write each expression with a common denominator of $x (- 2 y - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.5.3.8.1

Multiply $\frac{- 4 y^{2} - 2 y x - x}{- 2 y - x}$ by $\frac{x}{x}$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{(- 4 y^{2} - 2 y x - x) x}{(- 2 y - x) x}$

Step 5.4.5.3.8.2

Reorder the factors of $(- 2 y - x) x$ .

$x' = - \frac{2 y^{3}}{x (- 2 y - x)} + \frac{(- 4 y^{2} - 2 y x - x) x}{x (- 2 y - x)}$

Step 5.4.5.3.9

Combine the numerators over the common denominator.

$x' = \frac{- 2 y^{3} + (- 4 y^{2} - 2 y x - x) x}{x (- 2 y - x)}$

Step 5.4.5.3.10

Simplify the numerator.

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Step 5.4.5.3.10.1

Apply the distributive property.

$x' = \frac{- 2 y^{3} - 4 y^{2} x - 2 y x \cdot x - x \cdot x}{x (- 2 y - x)}$

Step 5.4.5.3.10.2

Simplify.

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Step 5.4.5.3.10.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.5.3.10.2.1.1

Move $x$ .

$x' = \frac{- 2 y^{3} - 4 y^{2} x - 2 y (x \cdot x) - x \cdot x}{x (- 2 y - x)}$

Step 5.4.5.3.10.2.1.2

Multiply $x$ by $x$ .

$x' = \frac{- 2 y^{3} - 4 y^{2} x - 2 y x^{2} - x \cdot x}{x (- 2 y - x)}$

Step 5.4.5.3.10.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.5.3.10.2.2.1

Move $x$ .

$x' = \frac{- 2 y^{3} - 4 y^{2} x - 2 y x^{2} - (x \cdot x)}{x (- 2 y - x)}$

Step 5.4.5.3.10.2.2.2

Multiply $x$ by $x$ .

$x' = \frac{- 2 y^{3} - 4 y^{2} x - 2 y x^{2} - x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.11

Simplify terms.

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Step 5.4.5.3.11.1

Factor $- 1$ out of $- 2 y^{3}$ .

$x' = \frac{- (2 y^{3}) - 4 y^{2} x - 2 y x^{2} - x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.11.2

Factor $- 1$ out of $- 4 y^{2} x$ .

$x' = \frac{- (2 y^{3}) - (4 y^{2} x) - 2 y x^{2} - x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.11.3

Factor $- 1$ out of $- (2 y^{3}) - (4 y^{2} x)$ .

$x' = \frac{- (2 y^{3} + 4 y^{2} x) - 2 y x^{2} - x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.11.4

Factor $- 1$ out of $- 2 y x^{2}$ .

$x' = \frac{- (2 y^{3} + 4 y^{2} x) - (2 y x^{2}) - x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.11.5

Factor $- 1$ out of $- (2 y^{3} + 4 y^{2} x) - (2 y x^{2})$ .

$x' = \frac{- (2 y^{3} + 4 y^{2} x + 2 y x^{2}) - x^{2}}{x (- 2 y - x)}$

Step 5.4.5.3.11.6

Factor $- 1$ out of $- x^{2}$ .

$x' = \frac{- (2 y^{3} + 4 y^{2} x + 2 y x^{2}) - (x^{2})}{x (- 2 y - x)}$

Step 5.4.5.3.11.7

Factor $- 1$ out of $- (2 y^{3} + 4 y^{2} x + 2 y x^{2}) - (x^{2})$ .

$x' = \frac{- (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})}{x (- 2 y - x)}$

Step 5.4.5.3.11.8

Rewrite $- (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})$ as $- 1 (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})$ .

$x' = \frac{- 1 (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})}{x (- 2 y - x)}$

Step 5.4.5.3.11.9

Factor $- 1$ out of $- 2 y$ .

$x' = \frac{- 1 (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})}{x (- (2 y) - x)}$

Step 5.4.5.3.11.10

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

$x' = \frac{- 1 (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})}{x (- (2 y) - (x))}$

Step 5.4.5.3.11.11

Factor $- 1$ out of $- (2 y) - (x)$ .

$x' = \frac{- 1 (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})}{x (- (2 y + x))}$

Step 5.4.5.3.11.12

Rewrite $- (2 y + x)$ as $- 1 (2 y + x)$ .

$x' = \frac{- 1 (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})}{x (- 1 (2 y + x))}$

Step 5.4.5.3.11.13

Cancel the common factor.

$x' = \frac{- 1 (2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2})}{x (- 1 (2 y + x))}$

Step 5.4.5.3.11.14

Rewrite the expression.

$x' = \frac{2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2}}{x (2 y + x)}$

Step 6

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

$\frac{d x}{d y} = \frac{2 y^{3} + 4 y^{2} x + 2 y x^{2} + x^{2}}{x (2 y + x)}$