Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $- 1$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $y$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Combine terms.

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Step 2.4.1.1

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

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

Step 2.4.1.2

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

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

Step 2.4.1.3

Write each expression with a common denominator of $x^{2} y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1.3.1

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

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

Step 2.4.1.3.2

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

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

Step 2.4.1.3.3

Reorder the factors of $y^{2} x^{2}$ .

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

Step 2.4.1.4

Combine the numerators over the common denominator.

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

Step 2.4.2

Reorder terms.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 y$ with respect to $x$ is $2 \frac{d}{d x} [y]$ .

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

Step 3.2

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

$2 y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

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

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Step 5.2.1.1.1

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

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Step 5.2.1.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.1.2

Rewrite the expression.

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

Step 5.2.1.1.2

Simplify each term.

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Step 5.2.1.1.2.1

Apply the distributive property.

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

Step 5.2.1.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.1.2.3

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

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Step 5.2.1.1.2.3.1

Move $y$ .

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

Step 5.2.1.1.2.3.2

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

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Step 5.2.1.1.2.3.2.1

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

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

Step 5.2.1.1.2.3.2.2

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

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

Step 5.2.1.1.2.3.3

Add $1$ and $2$ .

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

Step 5.2.1.1.2.4

Apply the distributive property.

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

Step 5.2.1.1.2.5

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.1.2.6

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

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Step 5.2.1.1.2.6.1

Move $x$ .

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

Step 5.2.1.1.2.6.2

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

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Step 5.2.1.1.2.6.2.1

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

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

Step 5.2.1.1.2.6.2.2

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

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

Step 5.2.1.1.2.6.3

Add $1$ and $2$ .

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

Step 5.2.1.1.3

Simplify the expression.

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Step 5.2.1.1.3.1

Move $y^{2}$ .

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

Step 5.2.1.1.3.2

Reorder $x$ and $y'$ .

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

Step 5.2.1.1.3.3

Move $x^{3}$ .

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

Step 5.2.1.1.3.4

Move $- y^{3}$ .

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

Step 5.2.1.1.3.5

Move $x^{2} y$ .

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

Step 5.2.1.1.3.6

Reorder $y' x y^{2}$ and $- y' x^{3}$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Remove parentheses.

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

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

$- y' x^{3} + y' x y^{2} + x^{2} y - y^{3} - 2 y' x^{2} y^{2} = 0$

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

Add $y^{3}$ to both sides of the equation.

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

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

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

Step 5.3.3.3

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

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

Step 5.3.3.4

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

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

Step 5.3.3.5

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

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

Step 5.3.4

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

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

Step 5.3.5

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

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Step 5.3.5.1

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

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

Step 5.3.5.2

Simplify the left side.

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Step 5.3.5.2.1

Cancel the common factor of $x$ .

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Step 5.3.5.2.1.1

Cancel the common factor.

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

Step 5.3.5.2.1.2

Rewrite the expression.

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

Step 5.3.5.2.2

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

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Step 5.3.5.2.2.1

Cancel the common factor.

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

Step 5.3.5.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.5.3

Simplify the right side.

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Step 5.3.5.3.1

Simplify each term.

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Step 5.3.5.3.1.1

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

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Step 5.3.5.3.1.1.1

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

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

Step 5.3.5.3.1.1.2

Cancel the common factors.

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Step 5.3.5.3.1.1.2.1

Cancel the common factor.

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

Step 5.3.5.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.5.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.5.3.2

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

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

Step 5.3.5.3.3

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

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Step 5.3.5.3.3.1

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

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

Step 5.3.5.3.3.2

Reorder the factors of $(- x^{2} + y^{2} - 2 x y^{2}) x$ .

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

Step 5.3.5.3.4

Combine the numerators over the common denominator.

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

Step 5.3.5.3.5

Simplify the numerator.

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Step 5.3.5.3.5.1

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

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Step 5.3.5.3.5.1.1

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

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

Step 5.3.5.3.5.1.2

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

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

Step 5.3.5.3.5.1.3

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

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

Step 5.3.5.3.5.2

Rewrite $x \cdot x$ as $x^{2}$ .

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

Step 5.3.5.3.5.3

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

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

Step 5.3.5.3.5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = x$ .

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

Step 5.3.5.3.6

Simplify with factoring out.

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Step 5.3.5.3.6.1

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

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

Step 5.3.5.3.6.2

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

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

Step 5.3.5.3.6.3

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

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

Step 5.3.5.3.6.4

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

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

Step 5.3.5.3.6.5

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

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

Step 5.3.5.3.6.6

Rewrite negatives.

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Step 5.3.5.3.6.6.1

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

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

Step 5.3.5.3.6.6.2

Move the negative in front of the fraction.

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

Step 6

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

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