Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ 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 2.2

Differentiate.

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Step 2.2.1

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 2.2.2

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 2.3

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 2.4

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 2.5

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 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Simplify the numerator.

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Step 2.8.2.1

Simplify each term.

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Step 2.8.2.1.1

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

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Step 2.8.2.1.1.1

Apply the distributive property.

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

Step 2.8.2.1.1.2

Apply the distributive property.

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

Step 2.8.2.1.1.3

Apply the distributive property.

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

Step 2.8.2.1.2

Simplify each term.

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Step 2.8.2.1.2.1

Multiply $x$ by $1$ .

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

Step 2.8.2.1.2.2

Multiply $- 1$ by $1$ .

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

Step 2.8.2.1.3

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

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Step 2.8.2.1.3.1

Apply the distributive property.

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

Step 2.8.2.1.3.2

Apply the distributive property.

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

Step 2.8.2.1.3.3

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 2.8.2.1.4

Simplify each term.

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Step 2.8.2.1.4.1

Multiply $- 1$ by $1$ .

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

Step 2.8.2.1.4.2

Multiply $- x (- y')$ .

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Step 2.8.2.1.4.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.8.2.1.4.2.2

Multiply $x$ by $1$ .

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

Step 2.8.2.1.4.3

Multiply $- 1$ by $1$ .

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

Step 2.8.2.1.4.4

Multiply $- y (- y')$ .

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Step 2.8.2.1.4.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.8.2.1.4.4.2

Multiply $y$ by $1$ .

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

Step 2.8.2.2

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

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Step 2.8.2.2.1

Subtract $x$ from $x$ .

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

Step 2.8.2.2.2

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

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

Step 2.8.2.2.3

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

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

Step 2.8.2.2.4

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

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

Step 2.8.2.3

Add $x y'$ and $x y'$ .

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

Step 2.8.2.4

Subtract $y$ from $- y$ .

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

Step 2.8.3

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

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Step 2.8.3.1

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

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

Step 2.8.3.2

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

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

Step 2.8.3.3

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

Evaluate $\frac{d}{d x} [y^{2}]$ .

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

$2 x + 2 u \frac{d}{d x} [y]$

Step 3.2.1.3

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

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

Step 3.2.2

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

$2 x + 2 y y'$

Step 3.3

Reorder terms.

$2 y y' + 2 x$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.1.2

Rewrite the expression.

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

Step 5.2.1.1.2

Apply the distributive property.

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

Step 5.2.1.1.3

Simplify the expression.

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Step 5.2.1.1.3.1

Multiply $- 1$ by $2$ .

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

Step 5.2.1.1.3.2

Move $x$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Move $y$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

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

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

Step 5.3.1.2

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

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Step 5.3.1.2.1

Apply the distributive property.

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

Step 5.3.1.2.2

Apply the distributive property.

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

Step 5.3.1.2.3

Apply the distributive property.

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

Step 5.3.1.3

Simplify and combine like terms.

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Step 5.3.1.3.1

Simplify each term.

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Step 5.3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 5.3.1.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.3.1.4

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

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Step 5.3.1.3.1.4.1

Move $y$ .

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

Step 5.3.1.3.1.4.2

Multiply $y$ by $y$ .

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

Step 5.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.3.1.3.1.6

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

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

Step 5.3.1.3.2

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

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Step 5.3.1.3.2.1

Move $y$ .

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

Step 5.3.1.3.2.2

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

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

Step 5.3.1.4

Apply the distributive property.

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

Step 5.3.1.5

Simplify.

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Step 5.3.1.5.1

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

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Step 5.3.1.5.1.1

Move $y$ .

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

Step 5.3.1.5.1.2

Multiply $y$ by $y$ .

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

Step 5.3.1.5.2

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

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Step 5.3.1.5.2.1

Move $y^{2}$ .

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

Step 5.3.1.5.2.2

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

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Step 5.3.1.5.2.2.1

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

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

Step 5.3.1.5.2.2.2

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

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

Step 5.3.1.5.2.3

Add $2$ and $1$ .

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

Step 5.3.1.6

Multiply $- 2$ by $2$ .

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

Step 5.3.1.7

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

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

Step 5.3.1.8

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

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Step 5.3.1.8.1

Apply the distributive property.

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

Step 5.3.1.8.2

Apply the distributive property.

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

Step 5.3.1.8.3

Apply the distributive property.

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

Step 5.3.1.9

Simplify and combine like terms.

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Step 5.3.1.9.1

Simplify each term.

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Step 5.3.1.9.1.1

Multiply $x$ by $x$ .

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

Step 5.3.1.9.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.9.1.3

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.9.1.4

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

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Step 5.3.1.9.1.4.1

Move $y$ .

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

Step 5.3.1.9.1.4.2

Multiply $y$ by $y$ .

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

Step 5.3.1.9.1.5

Multiply $- 1$ by $- 1$ .

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

Step 5.3.1.9.1.6

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

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

Step 5.3.1.9.2

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

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Step 5.3.1.9.2.1

Move $y$ .

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

Step 5.3.1.9.2.2

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

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

Step 5.3.1.10

Apply the distributive property.

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

Step 5.3.1.11

Simplify.

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Step 5.3.1.11.1

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

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Step 5.3.1.11.1.1

Move $x^{2}$ .

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

Step 5.3.1.11.1.2

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

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Step 5.3.1.11.1.2.1

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

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

Step 5.3.1.11.1.2.2

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

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

Step 5.3.1.11.1.3

Add $2$ and $1$ .

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

Step 5.3.1.11.2

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

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Step 5.3.1.11.2.1

Move $x$ .

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

Step 5.3.1.11.2.2

Multiply $x$ by $x$ .

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

Step 5.3.1.12

Simplify each term.

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Step 5.3.1.12.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.12.2

Multiply $2$ by $- 2$ .

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

Step 5.3.2

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 5.3.3

Subtract $2 y' x$ from both sides of the equation.

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

Step 5.3.4

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

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Step 5.3.4.1

Subtract $2 x^{3}$ from both sides of the equation.

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

Step 5.3.4.2

Add $4 x^{2} y$ to both sides of the equation.

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

Step 5.3.4.3

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

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

Step 5.3.5

Factor $2 y'$ out of $2 y' y x^{2} - 4 y' y^{2} x + 2 y' y^{3} - 2 y' x$ .

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Step 5.3.5.1

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

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

Step 5.3.5.2

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

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

Step 5.3.5.3

Factor $2 y'$ out of $2 y' y^{3}$ .

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

Step 5.3.5.4

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

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

Step 5.3.5.5

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

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

Step 5.3.5.6

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

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

Step 5.3.5.7

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

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

Step 5.3.6

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

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

Step 5.3.7

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

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Step 5.3.7.1

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

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

Step 5.3.7.2

Simplify the left side.

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Step 5.3.7.2.1

Cancel the common factor of $2$ .

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Step 5.3.7.2.1.1

Cancel the common factor.

Step 5.3.7.2.1.2

Rewrite the expression.

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

Step 5.3.7.2.2

Cancel the common factor of $y x^{2} - 2 y^{2} x + y^{3} - x$ .

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Step 5.3.7.2.2.1

Cancel the common factor.

Step 5.3.7.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.7.3

Simplify the right side.

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Step 5.3.7.3.1

Simplify each term.

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Step 5.3.7.3.1.1

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

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Step 5.3.7.3.1.1.1

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

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

Step 5.3.7.3.1.1.2

Cancel the common factors.

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Step 5.3.7.3.1.1.2.1

Cancel the common factor.

Step 5.3.7.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.7.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.7.3.1.3

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

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Step 5.3.7.3.1.3.1

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

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

Step 5.3.7.3.1.3.2

Cancel the common factors.

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Step 5.3.7.3.1.3.2.1

Cancel the common factor.

Step 5.3.7.3.1.3.2.2

Rewrite the expression.

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

Step 5.3.7.3.1.4

Move the negative in front of the fraction.

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

Step 5.3.7.3.1.5

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

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Step 5.3.7.3.1.5.1

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

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

Step 5.3.7.3.1.5.2

Cancel the common factors.

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Step 5.3.7.3.1.5.2.1

Cancel the common factor.

Step 5.3.7.3.1.5.2.2

Rewrite the expression.

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

Step 5.3.7.3.1.6

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

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Step 5.3.7.3.1.6.1

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

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

Step 5.3.7.3.1.6.2

Cancel the common factors.

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Step 5.3.7.3.1.6.2.1

Cancel the common factor.

Step 5.3.7.3.1.6.2.2

Rewrite the expression.

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

Step 5.3.7.3.1.7

Move the negative in front of the fraction.

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

Step 5.3.7.3.2

Simplify terms.

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Step 5.3.7.3.2.1

Combine the numerators over the common denominator.

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

Step 5.3.7.3.2.2

Combine the numerators over the common denominator.

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

Step 5.3.7.3.2.3

Combine the numerators over the common denominator.

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

Step 5.3.7.3.2.4

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

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

Step 5.3.7.3.2.5

Factor $- 1$ out of $- x^{3}$ .

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

Step 5.3.7.3.2.6

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

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

Step 5.3.7.3.2.7

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

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

Step 5.3.7.3.2.8

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

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

Step 5.3.7.3.2.9

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

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

Step 5.3.7.3.2.10

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

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

Step 5.3.7.3.2.11

Simplify the expression.

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Step 5.3.7.3.2.11.1

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

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

Step 5.3.7.3.2.11.2

Move the negative in front of the fraction.

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

Step 6

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

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