Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $4$ is constant with respect to $x$ , the derivative of $4 x^{2} y$ with respect to $x$ is $4 \frac{d}{d x} [x^{2} y]$ .

$4 \frac{d}{d x} [x^{2} y]$

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 $f (x) = x^{2}$ and $g (x) = y$ .

$4 (x^{2} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{2}])$

Step 3.3

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

$4 (x^{2} y' + y \frac{d}{d x} [x^{2}])$

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Multiply $2$ by $4$ .

$4 x^{2} y' + 8 y x$

Step 3.6.3

Reorder terms.

$4 x^{2} y' + 8 x y$

Step 4

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

$(2 x + 2 y y') (2 x^{2} + 2 y^{2}) = 4 x^{2} y' + 8 x y$

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.

$0 + 0 + (2 x + 2 y y') (2 x^{2} + 2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (2 x^{2} + 2 y^{2}) = 4 x^{2} y' + 8 x y$

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.

$2 x (2 x^{2} + 2 y^{2}) + 2 y y' (2 x^{2} + 2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.3.2

Apply the distributive property.

$2 x (2 x^{2}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2} + 2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.3.3

Apply the distributive property.

$2 x (2 x^{2}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.4

Simplify each term.

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Step 5.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x^{2} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

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 $x^{2}$ .

$2 \cdot 2 (x^{2} x) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

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 $1$ .

$2 \cdot 2 (x^{2} x^{1}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.4.2.2.2

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

$2 \cdot 2 x^{2 + 1} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.4.2.3

Add $2$ and $1$ .

$2 \cdot 2 x^{3} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.4.3

Multiply $2$ by $2$ .

$4 x^{3} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.4.4

Rewrite using the commutative property of multiplication.

$4 x^{3} + 2 \cdot 2 x y^{2} + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.4.5

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

Step 5.1.4.6

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y y' (2 y^{2}) = 4 x^{2} y' + 8 x y$

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 $y^{2}$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 (y^{2} y) y' \cdot 2 = 4 x^{2} y' + 8 x y$

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 $1$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 2 = 4 x^{2} y' + 8 x y$

Step 5.1.4.7.2.2

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

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y^{2 + 1} y' \cdot 2 = 4 x^{2} y' + 8 x y$

Step 5.1.4.7.3

Add $2$ and $1$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y^{3} y' \cdot 2 = 4 x^{2} y' + 8 x y$

Step 5.1.4.8

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' = 4 x^{2} y' + 8 x y$

Step 5.2

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

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' - 4 x^{2} y' = 8 x y$

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.

$4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' - 4 x^{2} y' = 8 x y - 4 x^{3}$

Step 5.3.2

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

$4 y y' x^{2} + 4 y^{3} y' - 4 x^{2} y' = 8 x y - 4 x^{3} - 4 x y^{2}$

Step 5.4

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

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Step 5.4.1

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

$4 y' (y x^{2}) + 4 y^{3} y' - 4 x^{2} y' = 8 x y - 4 x^{3} - 4 x y^{2}$

Step 5.4.2

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

$4 y' (y x^{2}) + 4 y' y^{3} - 4 x^{2} y' = 8 x y - 4 x^{3} - 4 x y^{2}$

Step 5.4.3

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

$4 y' (y x^{2}) + 4 y' y^{3} + 4 y' (- x^{2}) = 8 x y - 4 x^{3} - 4 x y^{2}$

Step 5.4.4

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

$4 y' (y x^{2} + y^{3}) + 4 y' (- x^{2}) = 8 x y - 4 x^{3} - 4 x y^{2}$

Step 5.4.5

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

$4 y' (y x^{2} + y^{3} - x^{2}) = 8 x y - 4 x^{3} - 4 x y^{2}$

Step 5.5

Divide each term in $4 y' (y x^{2} + y^{3} - x^{2}) = 8 x y - 4 x^{3} - 4 x y^{2}$ by $4 (y x^{2} + y^{3} - x^{2})$ and simplify.

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Step 5.5.1

Divide each term in $4 y' (y x^{2} + y^{3} - x^{2}) = 8 x y - 4 x^{3} - 4 x y^{2}$ by $4 (y x^{2} + y^{3} - x^{2})$ .

$\frac{4 y' (y x^{2} + y^{3} - x^{2})}{4 (y x^{2} + y^{3} - x^{2})} = \frac{8 x y}{4 (y x^{2} + y^{3} - x^{2})} + \frac{- 4 x^{3}}{4 (y x^{2} + y^{3} - x^{2})} + \frac{- 4 x y^{2}}{4 (y x^{2} + y^{3} - x^{2})}$

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.

$\frac{y' (y x^{2} + y^{3} - x^{2})}{y x^{2} + y^{3} - x^{2}} = \frac{8 x y}{4 (y x^{2} + y^{3} - x^{2})} + \frac{- 4 x^{3}}{4 (y x^{2} + y^{3} - x^{2})} + \frac{- 4 x y^{2}}{4 (y x^{2} + y^{3} - x^{2})}$

Step 5.5.2.2

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

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Step 5.5.2.2.1

Cancel the common factor.

Step 5.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{8 x y}{4 (y x^{2} + y^{3} - x^{2})} + \frac{- 4 x^{3}}{4 (y x^{2} + y^{3} - x^{2})} + \frac{- 4 x y^{2}}{4 (y x^{2} + y^{3} - x^{2})}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify terms.

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Step 5.5.3.1.1

Simplify each term.

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Step 5.5.3.1.1.1

Cancel the common factor of $8$ and $4$ .

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Step 5.5.3.1.1.1.1

Factor $4$ out of $8 x y$ .

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

Step 5.5.3.1.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.1.2.1

Cancel the common factor.

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

Step 5.5.3.1.1.1.2.2

Rewrite the expression.

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

Step 5.5.3.1.1.2

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

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Step 5.5.3.1.1.2.1

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

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

Step 5.5.3.1.1.2.2

Cancel the common factors.

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Step 5.5.3.1.1.2.2.1

Cancel the common factor.

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

Step 5.5.3.1.1.2.2.2

Rewrite the expression.

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

Step 5.5.3.1.1.3

Move the negative in front of the fraction.

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

Step 5.5.3.1.1.4

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

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Step 5.5.3.1.1.4.1

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

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

Step 5.5.3.1.1.4.2

Cancel the common factors.

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Step 5.5.3.1.1.4.2.1

Cancel the common factor.

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

Step 5.5.3.1.1.4.2.2

Rewrite the expression.

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

Step 5.5.3.1.1.5

Move the negative in front of the fraction.

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

Step 5.5.3.1.2

Combine into one fraction.

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Step 5.5.3.1.2.1

Combine the numerators over the common denominator.

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

Step 5.5.3.1.2.2

Combine the numerators over the common denominator.

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

Step 5.5.3.2

Simplify the numerator.

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Step 5.5.3.2.1

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

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Step 5.5.3.2.1.1

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

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

Step 5.5.3.2.1.2

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

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

Step 5.5.3.2.1.3

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

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

Step 5.5.3.2.1.4

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

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

Step 5.5.3.2.1.5

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

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

Step 5.5.3.2.2

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

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

Step 6

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

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