Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$2 x$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

$\frac{(x + y) (1 + \frac{d}{d x} [- y]) - (x - y) \frac{d}{d x} [x + y]}{{(x + y)}^{2}}$

Step 3.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$\frac{(x + y) (1 - \frac{d}{d x} [y]) - (x - y) \frac{d}{d x} [x + y]}{{(x + y)}^{2}}$

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Simplify the numerator.

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Step 3.7.2.1

Simplify each term.

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Step 3.7.2.1.1

Expand $(x + y) (1 - y')$ using the FOIL Method.

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Step 3.7.2.1.1.1

Apply the distributive property.

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

Step 3.7.2.1.1.2

Apply the distributive property.

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

Step 3.7.2.1.1.3

Apply the distributive property.

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

Step 3.7.2.1.2

Simplify each term.

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Step 3.7.2.1.2.1

Multiply $x$ by $1$ .

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

Step 3.7.2.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.7.2.1.2.3

Multiply $y$ by $1$ .

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

Step 3.7.2.1.2.4

Rewrite using the commutative property of multiplication.

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

Step 3.7.2.1.3

Multiply $- - y$ .

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Step 3.7.2.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.7.2.1.3.2

Multiply $y$ by $1$ .

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

Step 3.7.2.1.4

Expand $(- x + y) (1 + y')$ using the FOIL Method.

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Step 3.7.2.1.4.1

Apply the distributive property.

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

Step 3.7.2.1.4.2

Apply the distributive property.

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

Step 3.7.2.1.4.3

Apply the distributive property.

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

Step 3.7.2.1.5

Simplify each term.

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Step 3.7.2.1.5.1

Multiply $- 1$ by $1$ .

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

Step 3.7.2.1.5.2

Multiply $y$ by $1$ .

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

Step 3.7.2.2

Combine the opposite terms in $x - x y' + y - y y' - x - x y' + y + y y'$ .

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Step 3.7.2.2.1

Subtract $x$ from $x$ .

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

Step 3.7.2.2.2

Add $- x y' + y - y y'$ and $0$ .

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

Step 3.7.2.2.3

Add $- y y'$ and $y y'$ .

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

Step 3.7.2.2.4

Add $- x y' + y - x y' + y$ and $0$ .

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

Step 3.7.2.3

Subtract $x y'$ from $- x y'$ .

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

Step 3.7.2.4

Add $y$ and $y$ .

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

Step 3.7.3

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

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Step 3.7.3.1

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

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

Step 3.7.3.2

Factor $2$ out of $2 y$ .

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

Step 3.7.3.3

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

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

Step 3.7.4

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

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

Step 3.7.5

Factor $- 1$ out of $y$ .

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

Step 3.7.6

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

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

Step 3.7.7

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

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

Step 3.7.8

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- \frac{2 (x y' - y)}{{(x + y)}^{2}} = 2 x$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

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

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

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

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

Step 5.2.3.2

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

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

Step 5.2.3.3

Multiply $2$ by $- 1$ .

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

Step 5.3

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

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

Step 5.4

Simplify the left side.

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Step 5.4.1

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

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Step 5.4.1.1

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

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Step 5.4.1.1.1

Cancel the common factor.

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

Step 5.4.1.1.2

Rewrite the expression.

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

Step 5.4.1.2

Apply the distributive property.

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

Step 5.4.1.3

Simplify the expression.

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Step 5.4.1.3.1

Multiply $- 1$ by $2$ .

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

Step 5.4.1.3.2

Move $x$ .

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

Step 5.5

Solve for $y'$ .

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Step 5.5.1

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

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Step 5.5.1.1

Rewrite.

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

Step 5.5.1.2

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

$2 y' x - 2 y = - 2 x ((x + y) (x + y))$

Step 5.5.1.3

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

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Step 5.5.1.3.1

Apply the distributive property.

$2 y' x - 2 y = - 2 x (x (x + y) + y (x + y))$

Step 5.5.1.3.2

Apply the distributive property.

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

Step 5.5.1.3.3

Apply the distributive property.

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

Step 5.5.1.4

Simplify and combine like terms.

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Step 5.5.1.4.1

Simplify each term.

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Step 5.5.1.4.1.1

Multiply $x$ by $x$ .

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

Step 5.5.1.4.1.2

Multiply $y$ by $y$ .

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

Step 5.5.1.4.2

Add $x y$ and $y x$ .

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Step 5.5.1.4.2.1

Reorder $y$ and $x$ .

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

Step 5.5.1.4.2.2

Add $x y$ and $x y$ .

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

Step 5.5.1.5

Apply the distributive property.

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

Step 5.5.1.6

Simplify.

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Step 5.5.1.6.1

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

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Step 5.5.1.6.1.1

Move $x^{2}$ .

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

Step 5.5.1.6.1.2

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

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Step 5.5.1.6.1.2.1

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

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

Step 5.5.1.6.1.2.2

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

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

Step 5.5.1.6.1.3

Add $2$ and $1$ .

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

Step 5.5.1.6.2

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

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Step 5.5.1.6.2.1

Move $x$ .

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

Step 5.5.1.6.2.2

Multiply $x$ by $x$ .

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

Step 5.5.1.7

Simplify each term.

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Step 5.5.1.7.1

Rewrite using the commutative property of multiplication.

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

Step 5.5.1.7.2

Multiply $- 2$ by $2$ .

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

Step 5.5.2

Add $2 y$ to both sides of the equation.

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

Step 5.5.3

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

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Step 5.5.3.1

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

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

Step 5.5.3.2

Simplify the left side.

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Step 5.5.3.2.1

Cancel the common factor of $2$ .

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Step 5.5.3.2.1.1

Cancel the common factor.

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

Step 5.5.3.2.1.2

Rewrite the expression.

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

Step 5.5.3.2.2

Cancel the common factor of $x$ .

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Step 5.5.3.2.2.1

Cancel the common factor.

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

Step 5.5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.5.3.3

Simplify the right side.

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Step 5.5.3.3.1

Simplify each term.

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Step 5.5.3.3.1.1

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

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Step 5.5.3.3.1.1.1

Factor $2$ out of $- 2 x^{3}$ .

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

Step 5.5.3.3.1.1.2

Cancel the common factors.

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Step 5.5.3.3.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 5.5.3.3.1.1.2.2

Cancel the common factor.

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

Step 5.5.3.3.1.1.2.3

Rewrite the expression.

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

Step 5.5.3.3.1.2

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.5.3.3.1.2.1

Factor $x$ out of $- x^{3}$ .

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

Step 5.5.3.3.1.2.2

Cancel the common factors.

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Step 5.5.3.3.1.2.2.1

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

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

Step 5.5.3.3.1.2.2.2

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

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

Step 5.5.3.3.1.2.2.3

Cancel the common factor.

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

Step 5.5.3.3.1.2.2.4

Rewrite the expression.

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

Step 5.5.3.3.1.2.2.5

Divide $- x^{2}$ by $1$ .

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

Step 5.5.3.3.1.3

Cancel the common factor of $- 4$ and $2$ .

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Step 5.5.3.3.1.3.1

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

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

Step 5.5.3.3.1.3.2

Cancel the common factors.

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Step 5.5.3.3.1.3.2.1

Factor $2$ out of $2 x$ .

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

Step 5.5.3.3.1.3.2.2

Cancel the common factor.

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

Step 5.5.3.3.1.3.2.3

Rewrite the expression.

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

Step 5.5.3.3.1.4

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

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Step 5.5.3.3.1.4.1

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

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

Step 5.5.3.3.1.4.2

Cancel the common factors.

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Step 5.5.3.3.1.4.2.1

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

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

Step 5.5.3.3.1.4.2.2

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

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

Step 5.5.3.3.1.4.2.3

Cancel the common factor.

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

Step 5.5.3.3.1.4.2.4

Rewrite the expression.

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

Step 5.5.3.3.1.4.2.5

Divide $- 2 x y$ by $1$ .

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

Step 5.5.3.3.1.5

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

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Step 5.5.3.3.1.5.1

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

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

Step 5.5.3.3.1.5.2

Cancel the common factors.

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Step 5.5.3.3.1.5.2.1

Factor $2$ out of $2 x$ .

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

Step 5.5.3.3.1.5.2.2

Cancel the common factor.

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

Step 5.5.3.3.1.5.2.3

Rewrite the expression.

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

Step 5.5.3.3.1.6

Cancel the common factor of $x$ .

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Step 5.5.3.3.1.6.1

Cancel the common factor.

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

Step 5.5.3.3.1.6.2

Divide $- 1 y^{2}$ by $1$ .

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

Step 5.5.3.3.1.7

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

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

Step 5.5.3.3.1.8

Cancel the common factor of $2$ .

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Step 5.5.3.3.1.8.1

Cancel the common factor.

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

Step 5.5.3.3.1.8.2

Rewrite the expression.

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

Step 6

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

$\frac{d y}{d x} = - x^{2} - 2 x y - y^{2} + \frac{y}{x}$