Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 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 $g (x) = x^{2} + y^{2}$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [x^{2} + y^{2}]$

Step 2.1.2

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

$2 u_{1} \frac{d}{d x} [x^{2} + y^{2}]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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 2.3.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x y^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x y^{2}]$ .

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

Step 3.2

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

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

Step 3.3

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 3.3.1

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

$6 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 3.3.2

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

$6 (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 3.3.3

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

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

Step 3.4

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

$6 (2 \cdot x y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x])$

Step 3.5

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

$6 (2 x y y' + y^{2} \frac{d}{d x} [x])$

Step 3.6

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

$6 (2 x y y' + y^{2} \cdot 1)$

Step 3.7

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

$6 (2 x y y' + y^{2})$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$6 (2 x y y') + 6 y^{2}$

Step 3.8.2

Multiply $2$ by $6$ .

$12 x y y' + 6 y^{2}$

Step 3.8.3

Reorder terms.

$6 y^{2} + 12 x y y'$

Step 4

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

$(2 x + 2 y y') (2 x^{2} + 2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Rewrite.

$0 + 0 + (2 x + 2 y y') (2 x^{2} + 2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (2 x^{2} + 2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.3

Expand $(2 x + 2 y y') (2 x^{2} + 2 y^{2})$ using the FOIL Method.

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Step 5.1.3.1

Apply the distributive property.

$2 x (2 x^{2} + 2 y^{2}) + 2 y y' (2 x^{2} + 2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.3.2

Apply the distributive property.

$2 x (2 x^{2}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2} + 2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.3.3

Apply the distributive property.

$2 x (2 x^{2}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4

Simplify each term.

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Step 5.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x^{2} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.2

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

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Step 5.1.4.2.1

Move $x^{2}$ .

$2 \cdot 2 (x^{2} x) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.2.2

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

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Step 5.1.4.2.2.1

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

$2 \cdot 2 (x^{2} x^{1}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.2.2.2

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

$2 \cdot 2 x^{2 + 1} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.2.3

Add $2$ and $1$ .

$2 \cdot 2 x^{3} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.3

Multiply $2$ by $2$ .

$4 x^{3} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.4

Rewrite using the commutative property of multiplication.

$4 x^{3} + 2 \cdot 2 x y^{2} + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.5

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.6

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y y' (2 y^{2}) = 6 y^{2} + 12 x y y'$

Step 5.1.4.7

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

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Step 5.1.4.7.1

Move $y^{2}$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 (y^{2} y) y' \cdot 2 = 6 y^{2} + 12 x y y'$

Step 5.1.4.7.2

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

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Step 5.1.4.7.2.1

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

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 2 = 6 y^{2} + 12 x y y'$

Step 5.1.4.7.2.2

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

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y^{2 + 1} y' \cdot 2 = 6 y^{2} + 12 x y y'$

Step 5.1.4.7.3

Add $2$ and $1$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y^{3} y' \cdot 2 = 6 y^{2} + 12 x y y'$

Step 5.1.4.8

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' = 6 y^{2} + 12 x y y'$

Step 5.2

Subtract $12 x y y'$ from both sides of the equation.

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' - 12 x y y' = 6 y^{2}$

Step 5.3

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

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Step 5.3.1

Subtract $4 x^{3}$ from both sides of the equation.

$4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' - 12 x y y' = 6 y^{2} - 4 x^{3}$

Step 5.3.2

Subtract $4 x y^{2}$ from both sides of the equation.

$4 y y' x^{2} + 4 y^{3} y' - 12 x y y' = 6 y^{2} - 4 x^{3} - 4 x y^{2}$

Step 5.4

Factor $4 y y'$ out of $4 y y' x^{2} + 4 y^{3} y' - 12 x y y'$ .

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Step 5.4.1

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

$4 y y' x^{2} + 4 y^{3} y' - 12 x y y' = 6 y^{2} - 4 x^{3} - 4 x y^{2}$

Step 5.4.2

Factor $4 y y'$ out of $4 y^{3} y'$ .

$4 y y' x^{2} + 4 y y' y^{2} - 12 x y y' = 6 y^{2} - 4 x^{3} - 4 x y^{2}$

Step 5.4.3

Factor $4 y y'$ out of $- 12 x y y'$ .

$4 y y' x^{2} + 4 y y' y^{2} + 4 y y' (- 3 x) = 6 y^{2} - 4 x^{3} - 4 x y^{2}$

Step 5.4.4

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

$4 y y' (x^{2} + y^{2}) + 4 y y' (- 3 x) = 6 y^{2} - 4 x^{3} - 4 x y^{2}$

Step 5.4.5

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

$4 y y' (x^{2} + y^{2} - 3 x) = 6 y^{2} - 4 x^{3} - 4 x y^{2}$

Step 5.5

Divide each term in $4 y y' (x^{2} + y^{2} - 3 x) = 6 y^{2} - 4 x^{3} - 4 x y^{2}$ by $4 y (x^{2} + y^{2} - 3 x)$ and simplify.

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Step 5.5.1

Divide each term in $4 y y' (x^{2} + y^{2} - 3 x) = 6 y^{2} - 4 x^{3} - 4 x y^{2}$ by $4 y (x^{2} + y^{2} - 3 x)$ .

$\frac{4 y y' (x^{2} + y^{2} - 3 x)}{4 y (x^{2} + y^{2} - 3 x)} = \frac{6 y^{2}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x^{3}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x y^{2}}{4 y (x^{2} + y^{2} - 3 x)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $4$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y y' (x^{2} + y^{2} - 3 x)}{y (x^{2} + y^{2} - 3 x)} = \frac{6 y^{2}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x^{3}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x y^{2}}{4 y (x^{2} + y^{2} - 3 x)}$

Step 5.5.2.2

Cancel the common factor of $y$ .

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{y y' (x^{2} + y^{2} - 3 x)}{y (x^{2} + y^{2} - 3 x)} = \frac{6 y^{2}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x^{3}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x y^{2}}{4 y (x^{2} + y^{2} - 3 x)}$

Step 5.5.2.2.2

Rewrite the expression.

$\frac{y' (x^{2} + y^{2} - 3 x)}{x^{2} + y^{2} - 3 x} = \frac{6 y^{2}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x^{3}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x y^{2}}{4 y (x^{2} + y^{2} - 3 x)}$

Step 5.5.2.3

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

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Step 5.5.2.3.1

Cancel the common factor.

$\frac{y' (x^{2} + y^{2} - 3 x)}{x^{2} + y^{2} - 3 x} = \frac{6 y^{2}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x^{3}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x y^{2}}{4 y (x^{2} + y^{2} - 3 x)}$

Step 5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{6 y^{2}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x^{3}}{4 y (x^{2} + y^{2} - 3 x)} + \frac{- 4 x y^{2}}{4 y (x^{2} + y^{2} - 3 x)}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $6$ and $4$ .

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Step 5.5.3.1.1.1

Factor $2$ out of $6 y^{2}$ .

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

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

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

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

Step 5.5.3.1.1.2.2

Cancel the common factor.

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

Step 5.5.3.1.1.2.3

Rewrite the expression.

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

Step 5.5.3.1.2

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

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Step 5.5.3.1.2.1

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

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

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

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

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

Step 5.5.3.1.2.2.2

Cancel the common factor.

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

Step 5.5.3.1.2.2.3

Rewrite the expression.

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

Step 5.5.3.1.3

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

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Step 5.5.3.1.3.1

Factor $4$ out of $- 4 x^{3}$ .

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

Step 5.5.3.1.3.2

Cancel the common factors.

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Step 5.5.3.1.3.2.1

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

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

Step 5.5.3.1.3.2.2

Cancel the common factor.

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

Step 5.5.3.1.3.2.3

Rewrite the expression.

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

Step 5.5.3.1.4

Move the negative in front of the fraction.

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

Step 5.5.3.1.5

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

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Step 5.5.3.1.5.1

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

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

Step 5.5.3.1.5.2

Cancel the common factors.

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Step 5.5.3.1.5.2.1

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

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

Step 5.5.3.1.5.2.2

Cancel the common factor.

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

Step 5.5.3.1.5.2.3

Rewrite the expression.

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

Step 5.5.3.1.6

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

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Step 5.5.3.1.6.1

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

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

Step 5.5.3.1.6.2

Cancel the common factors.

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Step 5.5.3.1.6.2.1

Cancel the common factor.

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

Step 5.5.3.1.6.2.2

Rewrite the expression.

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

Step 5.5.3.1.7

Move the negative in front of the fraction.

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

Step 5.5.3.2

To write $- \frac{x y}{x^{2} + y^{2} - 3 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.5.3.3

Write each expression with a common denominator of $2 (x^{2} + y^{2} - 3 x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.1

Multiply $\frac{x y}{x^{2} + y^{2} - 3 x}$ by $\frac{2}{2}$ .

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

Step 5.5.3.3.2

Reorder the factors of $(x^{2} + y^{2} - 3 x) \cdot 2$ .

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

Step 5.5.3.4

Combine the numerators over the common denominator.

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

Step 5.5.3.5

Simplify the numerator.

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Step 5.5.3.5.1

Factor $y$ out of $3 y - x y \cdot 2$ .

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Step 5.5.3.5.1.1

Factor $y$ out of $3 y$ .

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

Step 5.5.3.5.1.2

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

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

Step 5.5.3.5.1.3

Factor $y$ out of $y \cdot 3 + y (- x \cdot 2)$ .

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

Step 5.5.3.5.2

Multiply $2$ by $- 1$ .

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

Step 5.5.3.6

To write $- \frac{x^{3}}{y (x^{2} + y^{2} - 3 x)}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.5.3.7

To write $\frac{y (3 - 2 x)}{2 (x^{2} + y^{2} - 3 x)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 5.5.3.8

Write each expression with a common denominator of $y (x^{2} + y^{2} - 3 x) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.8.1

Multiply $\frac{x^{3}}{y (x^{2} + y^{2} - 3 x)}$ by $\frac{2}{2}$ .

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

Step 5.5.3.8.2

Multiply $\frac{y (3 - 2 x)}{2 (x^{2} + y^{2} - 3 x)}$ by $\frac{y}{y}$ .

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

Step 5.5.3.8.3

Reorder the factors of $y (x^{2} + y^{2} - 3 x) \cdot 2$ .

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

Step 5.5.3.8.4

Reorder the factors of $2 (x^{2} + y^{2} - 3 x) y$ .

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

Step 5.5.3.9

Combine the numerators over the common denominator.

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

Step 5.5.3.10

Simplify the numerator.

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Step 5.5.3.10.1

Multiply $2$ by $- 1$ .

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

Step 5.5.3.10.2

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

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Step 5.5.3.10.2.1

Move $y$ .

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

Step 5.5.3.10.2.2

Multiply $y$ by $y$ .

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

Step 5.5.3.10.3

Apply the distributive property.

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

Step 5.5.3.10.4

Move $3$ to the left of $y^{2}$ .

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

Step 5.5.3.10.5

Rewrite using the commutative property of multiplication.

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

Step 5.5.3.11

Simplify with factoring out.

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Step 5.5.3.11.1

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

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

Step 5.5.3.11.2

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

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

Step 5.5.3.11.3

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

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

Step 5.5.3.11.4

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

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

Step 5.5.3.11.5

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

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

Step 5.5.3.11.6

Simplify the expression.

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Step 5.5.3.11.6.1

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

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

Step 5.5.3.11.6.2

Move the negative in front of the fraction.

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

Step 6

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

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