Calculus Examples

${(x^{2} + y^{2})}^{2} = \frac{25}{4} \cdot (x y^{2})$

Step 1

Remove parentheses.

${(x^{2} + y^{2})}^{2} = \frac{25}{4} \cdot (x y^{2})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (\frac{25}{4} \cdot (x y^{2}))$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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 3.1.1

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 3.1.2

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 3.1.3

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 3.2

Differentiate.

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Step 3.2.1

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 3.2.2

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 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 $u_{2}$ as $y$ .

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

Step 3.3.2

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 3.3.3

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

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

Step 3.4

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

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

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

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

Step 3.5.2

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 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Combine $x$ and $\frac{25}{4}$ .

$\frac{d}{d x} [\frac{x \cdot 25}{4} \cdot y^{2}]$

Step 4.1.2

Combine fractions.

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Step 4.1.2.1

Combine $\frac{x \cdot 25}{4}$ and $y^{2}$ .

$\frac{d}{d x} [\frac{x \cdot 25 y^{2}}{4}]$

Step 4.1.2.2

Move $25$ to the left of $x$ .

$\frac{d}{d x} [\frac{25 \cdot x y^{2}}{4}]$

Step 4.1.3

Since $\frac{25}{4}$ is constant with respect to $x$ , the derivative of $\frac{25 x y^{2}}{4}$ with respect to $x$ is $\frac{25}{4} \frac{d}{d x} [x y^{2}]$ .

$\frac{25}{4} \frac{d}{d x} [x y^{2}]$

Step 4.2

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

$\frac{25}{4} (x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x])$

Step 4.3

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 4.3.1

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

$\frac{25}{4} (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 4.3.2

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

$\frac{25}{4} (x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 4.3.3

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

$\frac{25}{4} (x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x])$

Step 4.4

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

$\frac{25}{4} (2 \cdot x y \frac{d}{d x} [y] + y^{2} \frac{d}{d x} [x])$

Step 4.5

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

$\frac{25}{4} (2 x y y' + y^{2} \frac{d}{d x} [x])$

Step 4.6

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

$\frac{25}{4} (2 x y y' + y^{2} \cdot 1)$

Step 4.7

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

$\frac{25}{4} (2 x y y' + y^{2})$

Step 4.8

Simplify.

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Step 4.8.1

Apply the distributive property.

$\frac{25}{4} (2 x y y') + \frac{25}{4} y^{2}$

Step 4.8.2

Combine terms.

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Step 4.8.2.1

Combine $2$ and $\frac{25}{4}$ .

$\frac{2 \cdot 25}{4} (x y y') + \frac{25}{4} y^{2}$

Step 4.8.2.2

Multiply $2$ by $25$ .

$\frac{50}{4} (x y y') + \frac{25}{4} y^{2}$

Step 4.8.2.3

Combine $x$ and $\frac{50}{4}$ .

$\frac{x \cdot 50}{4} (y y') + \frac{25}{4} y^{2}$

Step 4.8.2.4

Combine $y$ and $\frac{x \cdot 50}{4}$ .

$\frac{y (x \cdot 50)}{4} y' + \frac{25}{4} y^{2}$

Step 4.8.2.5

Combine $\frac{y (x \cdot 50)}{4}$ and $y'$ .

$\frac{y (x \cdot 50) y'}{4} + \frac{25}{4} y^{2}$

Step 4.8.2.6

Move $50$ to the left of $x$ .

$\frac{y (50 \cdot x) y'}{4} + \frac{25}{4} y^{2}$

Step 4.8.2.7

Move $50$ to the left of $y$ .

$\frac{50 \cdot y x y'}{4} + \frac{25}{4} y^{2}$

Step 4.8.2.8

Cancel the common factor of $50$ and $4$ .

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Step 4.8.2.8.1

Factor $2$ out of $50 y x y'$ .

$\frac{2 (25 y x y')}{4} + \frac{25}{4} y^{2}$

Step 4.8.2.8.2

Cancel the common factors.

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Step 4.8.2.8.2.1

Factor $2$ out of $4$ .

$\frac{2 (25 y x y')}{2 (2)} + \frac{25}{4} y^{2}$

Step 4.8.2.8.2.2

Cancel the common factor.

$\frac{2 (25 y x y')}{2 \cdot 2} + \frac{25}{4} y^{2}$

Step 4.8.2.8.2.3

Rewrite the expression.

$\frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 5

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

$(2 x + 2 y y') (2 x^{2} + 2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6

Solve for $y'$ .

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Step 6.1

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

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Step 6.1.1

Rewrite.

$0 + 0 + (2 x + 2 y y') (2 x^{2} + 2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (2 x^{2} + 2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.3

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

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Step 6.1.3.1

Apply the distributive property.

$2 x (2 x^{2} + 2 y^{2}) + 2 y y' (2 x^{2} + 2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.3.2

Apply the distributive property.

$2 x (2 x^{2}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2} + 2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.3.3

Apply the distributive property.

$2 x (2 x^{2}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4

Simplify each term.

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Step 6.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x^{2} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.2

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

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Step 6.1.4.2.1

Move $x^{2}$ .

$2 \cdot 2 (x^{2} x) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.2.2

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

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Step 6.1.4.2.2.1

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

$2 \cdot 2 (x^{2} x^{1}) + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.2.2.2

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

$2 \cdot 2 x^{2 + 1} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.2.3

Add $2$ and $1$ .

$2 \cdot 2 x^{3} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.3

Multiply $2$ by $2$ .

$4 x^{3} + 2 x (2 y^{2}) + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.4

Rewrite using the commutative property of multiplication.

$4 x^{3} + 2 \cdot 2 x y^{2} + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.5

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 2 y y' (2 x^{2}) + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.6

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y y' (2 y^{2}) = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.7

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

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Step 6.1.4.7.1

Move $y^{2}$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 (y^{2} y) y' \cdot 2 = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.7.2

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

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Step 6.1.4.7.2.1

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

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 2 = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.7.2.2

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

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y^{2 + 1} y' \cdot 2 = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.7.3

Add $2$ and $1$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 2 y^{3} y' \cdot 2 = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.1.4.8

Multiply $2$ by $2$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' = \frac{25 y x y'}{2} + \frac{25}{4} y^{2}$

Step 6.2

Combine $\frac{25}{4}$ and $y^{2}$ .

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' = \frac{25 y x y'}{2} + \frac{25 y^{2}}{4}$

Step 6.3

Subtract $\frac{25 y x y'}{2}$ from both sides of the equation.

$4 x^{3} + 4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' - \frac{25 y x y'}{2} = \frac{25 y^{2}}{4}$

Step 6.4

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

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Step 6.4.1

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

$4 x y^{2} + 4 y y' x^{2} + 4 y^{3} y' - \frac{25 y x y'}{2} = \frac{25 y^{2}}{4} - 4 x^{3}$

Step 6.4.2

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

$4 y y' x^{2} + 4 y^{3} y' - \frac{25 y x y'}{2} = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$

Step 6.5

Factor $y y'$ out of $4 y y' x^{2} + 4 y^{3} y' - \frac{25 y x y'}{2}$ .

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Step 6.5.1

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

$y y' (4 x^{2}) + 4 y^{3} y' - \frac{25 y x y'}{2} = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$

Step 6.5.2

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

$y y' (4 x^{2}) + y y' (4 y^{2}) - \frac{25 y x y'}{2} = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$

Step 6.5.3

Factor $y y'$ out of $- \frac{25 y x y'}{2}$ .

$y y' (4 x^{2}) + y y' (4 y^{2}) + y y' (- \frac{25 x}{2}) = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$

Step 6.5.4

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

$y y' (4 x^{2} + 4 y^{2}) + y y' (- \frac{25 x}{2}) = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$

Step 6.5.5

Factor $y y'$ out of $y y' (4 x^{2} + 4 y^{2}) + y y' (- \frac{25 x}{2})$ .

$y y' (4 x^{2} + 4 y^{2} - \frac{25 x}{2}) = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$

Step 6.6

Divide each term in $y y' (4 x^{2} + 4 y^{2} - \frac{25 x}{2}) = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$ by $y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})$ and simplify.

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Step 6.6.1

Divide each term in $y y' (4 x^{2} + 4 y^{2} - \frac{25 x}{2}) = \frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}$ by $y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})$ .

$\frac{y y' (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})} = \frac{\frac{25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})} + \frac{- 4 x^{3}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})} + \frac{- 4 x y^{2}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.2

Simplify the left side.

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Step 6.6.2.1

Cancel the common factor of $y$ .

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Step 6.6.2.1.1

Cancel the common factor.

Step 6.6.2.1.2

Rewrite the expression.

$\frac{y' (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}{4 x^{2} + 4 y^{2} - \frac{25 x}{2}} = \frac{\frac{25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})} + \frac{- 4 x^{3}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})} + \frac{- 4 x y^{2}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.2.2

Cancel the common factor of $4 x^{2} + 4 y^{2} - \frac{25 x}{2}$ .

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Step 6.6.2.2.1

Cancel the common factor.

Step 6.6.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})} + \frac{- 4 x^{3}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})} + \frac{- 4 x y^{2}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3

Simplify the right side.

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Step 6.6.3.1

Combine the numerators over the common denominator.

$y' = \frac{\frac{25 y^{2}}{4} - 4 x^{3} - 4 x y^{2}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.2

Reorder the factors of $- 4 x y^{2}$ .

$y' = \frac{\frac{25 y^{2}}{4} - 4 x^{3} - 4 y^{2} x}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.3

Subtract $4 y^{2} x$ from $\frac{25 y^{2}}{4}$ .

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Step 6.6.3.3.1

Reorder $\frac{25 y^{2}}{4}$ and $- 4 x y^{2}$ .

$y' = \frac{- 4 x^{3} - 4 x y^{2} + \frac{25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.3.2

To write $- 4 x y^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y' = \frac{- 4 x^{3} - 4 x y^{2} \cdot \frac{4}{4} + \frac{25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.3.3

Combine $- 4 x y^{2}$ and $\frac{4}{4}$ .

$y' = \frac{- 4 x^{3} + \frac{- 4 x y^{2} \cdot 4}{4} + \frac{25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.3.4

Combine the numerators over the common denominator.

$y' = \frac{- 4 x^{3} + \frac{- 4 x y^{2} \cdot 4 + 25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.4

Simplify the numerator.

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Step 6.6.3.4.1

Factor $y^{2}$ out of $- 4 x y^{2} \cdot 4 + 25 y^{2}$ .

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Step 6.6.3.4.1.1

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

$y' = \frac{- 4 x^{3} + \frac{y^{2} (- 4 x \cdot 4) + 25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.4.1.2

Factor $y^{2}$ out of $25 y^{2}$ .

$y' = \frac{- 4 x^{3} + \frac{y^{2} (- 4 x \cdot 4) + y^{2} \cdot 25}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.4.1.3

Factor $y^{2}$ out of $y^{2} (- 4 x \cdot 4) + y^{2} \cdot 25$ .

$y' = \frac{- 4 x^{3} + \frac{y^{2} (- 4 x \cdot 4 + 25)}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.4.2

Multiply $4$ by $- 4$ .

$y' = \frac{- 4 x^{3} + \frac{y^{2} (- 16 x + 25)}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.5

To write $- 4 x^{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y' = \frac{- 4 x^{3} \cdot \frac{4}{4} + \frac{y^{2} (- 16 x + 25)}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.6

Simplify terms.

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Step 6.6.3.6.1

Combine $- 4 x^{3}$ and $\frac{4}{4}$ .

$y' = \frac{\frac{- 4 x^{3} \cdot 4}{4} + \frac{y^{2} (- 16 x + 25)}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.6.2

Combine the numerators over the common denominator.

$y' = \frac{\frac{- 4 x^{3} \cdot 4 + y^{2} (- 16 x + 25)}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.7

Simplify the numerator.

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Step 6.6.3.7.1

Multiply $4$ by $- 4$ .

$y' = \frac{\frac{- 16 x^{3} + y^{2} (- 16 x + 25)}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.7.2

Apply the distributive property.

$y' = \frac{\frac{- 16 x^{3} + y^{2} (- 16 x) + y^{2} \cdot 25}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.7.3

Rewrite using the commutative property of multiplication.

$y' = \frac{\frac{- 16 x^{3} - 16 y^{2} x + y^{2} \cdot 25}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.7.4

Move $25$ to the left of $y^{2}$ .

$y' = \frac{\frac{- 16 x^{3} - 16 y^{2} x + 25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.8

Simplify with factoring out.

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Step 6.6.3.8.1

Factor $- 1$ out of $- 16 x^{3}$ .

$y' = \frac{\frac{- (16 x^{3}) - 16 y^{2} x + 25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.8.2

Factor $- 1$ out of $- 16 y^{2} x$ .

$y' = \frac{\frac{- (16 x^{3}) - (16 y^{2} x) + 25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.8.3

Factor $- 1$ out of $- (16 x^{3}) - (16 y^{2} x)$ .

$y' = \frac{\frac{- (16 x^{3} + 16 y^{2} x) + 25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.8.4

Factor $- 1$ out of $25 y^{2}$ .

$y' = \frac{\frac{- (16 x^{3} + 16 y^{2} x) - (- 25 y^{2})}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.8.5

Factor $- 1$ out of $- (16 x^{3} + 16 y^{2} x) - (- 25 y^{2})$ .

$y' = \frac{\frac{- (16 x^{3} + 16 y^{2} x - 25 y^{2})}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.8.6

Simplify the expression.

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Step 6.6.3.8.6.1

Rewrite $- (16 x^{3} + 16 y^{2} x - 25 y^{2})$ as $- 1 (16 x^{3} + 16 y^{2} x - 25 y^{2})$ .

$y' = \frac{\frac{- 1 (16 x^{3} + 16 y^{2} x - 25 y^{2})}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.8.6.2

Move the negative in front of the fraction.

$y' = \frac{- \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4}}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.9

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 x^{2} + 4 y^{2} - \frac{25 x}{2})}$

Step 6.6.3.10

Simplify the denominator.

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Step 6.6.3.10.1

To write $4 x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} + 4 x^{2} \cdot \frac{2}{2} - \frac{25 x}{2})}$

Step 6.6.3.10.2

Combine $4 x^{2}$ and $\frac{2}{2}$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} + \frac{4 x^{2} \cdot 2}{2} - \frac{25 x}{2})}$

Step 6.6.3.10.3

Combine the numerators over the common denominator.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} + \frac{4 x^{2} \cdot 2 - 25 x}{2})}$

Step 6.6.3.10.4

Simplify the numerator.

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Step 6.6.3.10.4.1

Factor $x$ out of $4 x^{2} \cdot 2 - 25 x$ .

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Step 6.6.3.10.4.1.1

Factor $x$ out of $4 x^{2} \cdot 2$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} + \frac{x (4 x \cdot 2) - 25 x}{2})}$

Step 6.6.3.10.4.1.2

Factor $x$ out of $- 25 x$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} + \frac{x (4 x \cdot 2) + x \cdot - 25}{2})}$

Step 6.6.3.10.4.1.3

Factor $x$ out of $x (4 x \cdot 2) + x \cdot - 25$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} + \frac{x (4 x \cdot 2 - 25)}{2})}$

Step 6.6.3.10.4.2

Multiply $2$ by $4$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} + \frac{x (8 x - 25)}{2})}$

Step 6.6.3.10.5

To write $4 y^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (4 y^{2} \cdot \frac{2}{2} + \frac{x (8 x - 25)}{2})}$

Step 6.6.3.10.6

Combine $4 y^{2}$ and $\frac{2}{2}$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y (\frac{4 y^{2} \cdot 2}{2} + \frac{x (8 x - 25)}{2})}$

Step 6.6.3.10.7

Combine the numerators over the common denominator.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y \frac{4 y^{2} \cdot 2 + x (8 x - 25)}{2}}$

Step 6.6.3.10.8

Simplify the numerator.

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Step 6.6.3.10.8.1

Multiply $2$ by $4$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y \frac{8 y^{2} + x (8 x - 25)}{2}}$

Step 6.6.3.10.8.2

Apply the distributive property.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y \frac{8 y^{2} + x (8 x) + x \cdot - 25}{2}}$

Step 6.6.3.10.8.3

Rewrite using the commutative property of multiplication.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y \frac{8 y^{2} + 8 x \cdot x + x \cdot - 25}{2}}$

Step 6.6.3.10.8.4

Move $- 25$ to the left of $x$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y \frac{8 y^{2} + 8 x \cdot x - 25 \cdot x}{2}}$

Step 6.6.3.10.8.5

Simplify each term.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{y \frac{8 y^{2} + 8 x^{2} - 25 x}{2}}$

Step 6.6.3.11

Combine $y$ and $\frac{8 y^{2} + 8 x^{2} - 25 x}{2}$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{1}{\frac{y (8 y^{2} + 8 x^{2} - 25 x)}{2}}$

Step 6.6.3.12

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} (1 \frac{2}{y (8 y^{2} + 8 x^{2} - 25 x)})$

Step 6.6.3.13

Multiply $\frac{2}{y (8 y^{2} + 8 x^{2} - 25 x)}$ by $1$ .

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4} \cdot \frac{2}{y (8 y^{2} + 8 x^{2} - 25 x)}$

Step 6.6.3.14

Cancel the common factor of $2$ .

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Step 6.6.3.14.1

Move the leading negative in $- \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{4}$ into the numerator.

$y' = \frac{- (16 x^{3} + 16 y^{2} x - 25 y^{2})}{4} \cdot \frac{2}{y (8 y^{2} + 8 x^{2} - 25 x)}$

Step 6.6.3.14.2

Factor $2$ out of $4$ .

$y' = \frac{- (16 x^{3} + 16 y^{2} x - 25 y^{2})}{2 (2)} \cdot \frac{2}{y (8 y^{2} + 8 x^{2} - 25 x)}$

Step 6.6.3.14.3

Cancel the common factor.

$y' = \frac{- (16 x^{3} + 16 y^{2} x - 25 y^{2})}{2 \cdot 2} \cdot \frac{2}{y (8 y^{2} + 8 x^{2} - 25 x)}$

Step 6.6.3.14.4

Rewrite the expression.

$y' = \frac{- (16 x^{3} + 16 y^{2} x - 25 y^{2})}{2} \cdot \frac{1}{y (8 y^{2} + 8 x^{2} - 25 x)}$

Step 6.6.3.15

Multiply $\frac{- (16 x^{3} + 16 y^{2} x - 25 y^{2})}{2}$ by $\frac{1}{y (8 y^{2} + 8 x^{2} - 25 x)}$ .

$y' = \frac{- (16 x^{3} + 16 y^{2} x - 25 y^{2})}{2 (y (8 y^{2} + 8 x^{2} - 25 x))}$

Step 6.6.3.16

Move the negative in front of the fraction.

$y' = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{2 y (8 y^{2} + 8 x^{2} - 25 x)}$

Step 7

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

$\frac{d y}{d x} = - \frac{16 x^{3} + 16 y^{2} x - 25 y^{2}}{2 y (8 y^{2} + 8 x^{2} - 25 x)}$