Calculus Examples

$2 {(x^{2} + y^{2})}^{2} = 25 (x^{2} - y^{2})$

Step 1

Set each solution of $2 {(x^{2} + y^{2})}^{2}$ as a function of $x$ .

$y = 25 (x^{2} - y^{2}) \to f (x) = 25 (x^{2} - y^{2})$

Step 2

Because the $y$ variable in the equation $2 {(x^{2} + y^{2})}^{2} = 25 (x^{2} - y^{2})$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 2.1

Differentiate both sides of the equation.

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

Step 2.2

Differentiate the left side of the equation.

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Step 2.2.1

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

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

Step 2.2.2

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 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

Differentiate.

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Step 2.2.3.1

Multiply $2$ by $2$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.4

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 2.2.4.1

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

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

Step 2.2.4.2

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

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

Step 2.2.4.3

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

Apply the distributive property.

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

Step 2.2.6.2

Reorder the factors of $(4 x^{2} + 4 y^{2}) (2 x + 2 y y')$ .

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

Step 2.3

Differentiate the right side of the equation.

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Step 2.3.1

Differentiate.

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{2}$ with respect to $x$ is $- \frac{d}{d x} [y^{2}]$ .

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

Step 2.3.2

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 2.3.2.1

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

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

Step 2.3.2.2

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

$25 (2 x - (2 u \frac{d}{d x} [y]))$

Step 2.3.2.3

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

$25 (2 x - (2 y \frac{d}{d x} [y]))$

Step 2.3.3

Multiply $2$ by $- 1$ .

$25 (2 x - 2 (y \frac{d}{d x} [y]))$

Step 2.3.4

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

$25 (2 x - 2 y y')$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Apply the distributive property.

$25 (2 x) + 25 (- 2 y y')$

Step 2.3.5.2

Combine terms.

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Step 2.3.5.2.1

Multiply $2$ by $25$ .

$50 x + 25 (- 2 y y')$

Step 2.3.5.2.2

Multiply $- 2$ by $25$ .

$50 x - 50 y y'$

Step 2.3.5.3

Reorder terms.

$- 50 y y' + 50 x$

Step 2.4

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

$(2 x + 2 y y') (4 x^{2} + 4 y^{2}) = - 50 y y' + 50 x$

Step 2.5

Solve for $y'$ .

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Step 2.5.1

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

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Step 2.5.1.1

Rewrite.

$0 + 0 + (2 x + 2 y y') (4 x^{2} + 4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.2

Simplify by adding zeros.

$(2 x + 2 y y') (4 x^{2} + 4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.3

Expand $(2 x + 2 y y') (4 x^{2} + 4 y^{2})$ using the FOIL Method.

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Step 2.5.1.3.1

Apply the distributive property.

$2 x (4 x^{2} + 4 y^{2}) + 2 y y' (4 x^{2} + 4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.3.2

Apply the distributive property.

$2 x (4 x^{2}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2} + 4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.3.3

Apply the distributive property.

$2 x (4 x^{2}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4

Simplify each term.

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Step 2.5.1.4.1

Rewrite using the commutative property of multiplication.

$2 \cdot 4 x \cdot x^{2} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.2

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

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Step 2.5.1.4.2.1

Move $x^{2}$ .

$2 \cdot 4 (x^{2} x) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.2.2

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

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Step 2.5.1.4.2.2.1

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

$2 \cdot 4 (x^{2} x^{1}) + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.2.2.2

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

$2 \cdot 4 x^{2 + 1} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.2.3

Add $2$ and $1$ .

$2 \cdot 4 x^{3} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.3

Multiply $2$ by $4$ .

$8 x^{3} + 2 x (4 y^{2}) + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.4

Rewrite using the commutative property of multiplication.

$8 x^{3} + 2 \cdot 4 x y^{2} + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.5

Multiply $2$ by $4$ .

$8 x^{3} + 8 x y^{2} + 2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.6

Multiply $4$ by $2$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y y' (4 y^{2}) = - 50 y y' + 50 x$

Step 2.5.1.4.7

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

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Step 2.5.1.4.7.1

Move $y^{2}$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 (y^{2} y) y' \cdot 4 = - 50 y y' + 50 x$

Step 2.5.1.4.7.2

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

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Step 2.5.1.4.7.2.1

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

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 (y^{2} y^{1}) y' \cdot 4 = - 50 y y' + 50 x$

Step 2.5.1.4.7.2.2

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

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y^{2 + 1} y' \cdot 4 = - 50 y y' + 50 x$

Step 2.5.1.4.7.3

Add $2$ and $1$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 2 y^{3} y' \cdot 4 = - 50 y y' + 50 x$

Step 2.5.1.4.8

Multiply $4$ by $2$ .

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' = - 50 y y' + 50 x$

Step 2.5.2

Add $50 y y'$ to both sides of the equation.

$8 x^{3} + 8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' + 50 y y' = 50 x$

Step 2.5.3

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

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Step 2.5.3.1

Subtract $8 x^{3}$ from both sides of the equation.

$8 x y^{2} + 8 y y' x^{2} + 8 y^{3} y' + 50 y y' = 50 x - 8 x^{3}$

Step 2.5.3.2

Subtract $8 x y^{2}$ from both sides of the equation.

$8 y y' x^{2} + 8 y^{3} y' + 50 y y' = 50 x - 8 x^{3} - 8 x y^{2}$

Step 2.5.4

Factor $2 y y'$ out of $8 y y' x^{2} + 8 y^{3} y' + 50 y y'$ .

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Step 2.5.4.1

Factor $2 y y'$ out of $8 y y' x^{2}$ .

$2 y y' (4 x^{2}) + 8 y^{3} y' + 50 y y' = 50 x - 8 x^{3} - 8 x y^{2}$

Step 2.5.4.2

Factor $2 y y'$ out of $8 y^{3} y'$ .

$2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) + 50 y y' = 50 x - 8 x^{3} - 8 x y^{2}$

Step 2.5.4.3

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

$2 y y' (4 x^{2}) + 2 y y' (4 y^{2}) + 2 y y' \cdot 25 = 50 x - 8 x^{3} - 8 x y^{2}$

Step 2.5.4.4

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

$2 y y' (4 x^{2} + 4 y^{2}) + 2 y y' \cdot 25 = 50 x - 8 x^{3} - 8 x y^{2}$

Step 2.5.4.5

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

$2 y y' (4 x^{2} + 4 y^{2} + 25) = 50 x - 8 x^{3} - 8 x y^{2}$

Step 2.5.5

Divide each term in $2 y y' (4 x^{2} + 4 y^{2} + 25) = 50 x - 8 x^{3} - 8 x y^{2}$ by $2 y (4 x^{2} + 4 y^{2} + 25)$ and simplify.

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Step 2.5.5.1

Divide each term in $2 y y' (4 x^{2} + 4 y^{2} + 25) = 50 x - 8 x^{3} - 8 x y^{2}$ by $2 y (4 x^{2} + 4 y^{2} + 25)$ .

$\frac{2 y y' (4 x^{2} + 4 y^{2} + 25)}{2 y (4 x^{2} + 4 y^{2} + 25)} = \frac{50 x}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} + 25)}$

Step 2.5.5.2

Simplify the left side.

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Step 2.5.5.2.1

Cancel the common factor of $2$ .

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Step 2.5.5.2.1.1

Cancel the common factor.

Step 2.5.5.2.1.2

Rewrite the expression.

$\frac{y y' (4 x^{2} + 4 y^{2} + 25)}{y (4 x^{2} + 4 y^{2} + 25)} = \frac{50 x}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} + 25)}$

Step 2.5.5.2.2

Cancel the common factor of $y$ .

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Step 2.5.5.2.2.1

Cancel the common factor.

Step 2.5.5.2.2.2

Rewrite the expression.

$\frac{y' (4 x^{2} + 4 y^{2} + 25)}{4 x^{2} + 4 y^{2} + 25} = \frac{50 x}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} + 25)}$

Step 2.5.5.2.3

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

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Step 2.5.5.2.3.1

Cancel the common factor.

Step 2.5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{50 x}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x^{3}}{2 y (4 x^{2} + 4 y^{2} + 25)} + \frac{- 8 x y^{2}}{2 y (4 x^{2} + 4 y^{2} + 25)}$

Step 2.5.5.3

Simplify the right side.

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Step 2.5.5.3.1

Simplify each term.

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Step 2.5.5.3.1.1

Cancel the common factor of $50$ and $2$ .

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Step 2.5.5.3.1.1.1

Factor $2$ out of $50 x$ .

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

Step 2.5.5.3.1.1.2

Cancel the common factors.

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Step 2.5.5.3.1.1.2.1

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

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

Step 2.5.5.3.1.1.2.2

Cancel the common factor.

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

Step 2.5.5.3.1.1.2.3

Rewrite the expression.

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

Step 2.5.5.3.1.2

Cancel the common factor of $- 8$ and $2$ .

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Step 2.5.5.3.1.2.1

Factor $2$ out of $- 8 x^{3}$ .

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

Step 2.5.5.3.1.2.2

Cancel the common factors.

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Step 2.5.5.3.1.2.2.1

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

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

Step 2.5.5.3.1.2.2.2

Cancel the common factor.

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

Step 2.5.5.3.1.2.2.3

Rewrite the expression.

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

Step 2.5.5.3.1.3

Move the negative in front of the fraction.

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

Step 2.5.5.3.1.4

Cancel the common factor of $- 8$ and $2$ .

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Step 2.5.5.3.1.4.1

Factor $2$ out of $- 8 x y^{2}$ .

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

Step 2.5.5.3.1.4.2

Cancel the common factors.

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Step 2.5.5.3.1.4.2.1

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

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

Step 2.5.5.3.1.4.2.2

Cancel the common factor.

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

Step 2.5.5.3.1.4.2.3

Rewrite the expression.

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

Step 2.5.5.3.1.5

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

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Step 2.5.5.3.1.5.1

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

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

Step 2.5.5.3.1.5.2

Cancel the common factors.

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Step 2.5.5.3.1.5.2.1

Cancel the common factor.

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

Step 2.5.5.3.1.5.2.2

Rewrite the expression.

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

Step 2.5.5.3.1.6

Move the negative in front of the fraction.

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

Step 2.5.5.3.2

To write $- \frac{4 x y}{4 x^{2} + 4 y^{2} + 25}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 2.5.5.3.3

Write each expression with a common denominator of $y (4 x^{2} + 4 y^{2} + 25)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.5.3.3.1

Multiply $\frac{4 x y}{4 x^{2} + 4 y^{2} + 25}$ by $\frac{y}{y}$ .

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

Step 2.5.5.3.3.2

Reorder the factors of $(4 x^{2} + 4 y^{2} + 25) y$ .

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

Step 2.5.5.3.4

Combine the numerators over the common denominator.

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

Step 2.5.5.3.5

Combine the numerators over the common denominator.

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

Step 2.5.5.3.6

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

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Step 2.5.5.3.6.1

Move $y$ .

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

Step 2.5.5.3.6.2

Multiply $y$ by $y$ .

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

Step 2.5.5.3.7

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

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Step 2.5.5.3.7.1

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

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

Step 2.5.5.3.7.2

Factor $x$ out of $25 x$ .

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

Step 2.5.5.3.7.3

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

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

Step 2.5.5.3.7.4

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

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

Step 2.5.5.3.7.5

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

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

Step 2.5.5.3.8

Factor $- 1$ out of $- 4 x^{2}$ .

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

Step 2.5.5.3.9

Rewrite $25$ as $- 1 (- 25)$ .

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

Step 2.5.5.3.10

Factor $- 1$ out of $- (4 x^{2}) - 1 (- 25)$ .

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

Step 2.5.5.3.11

Factor $- 1$ out of $- 4 y^{2}$ .

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

Step 2.5.5.3.12

Factor $- 1$ out of $- (4 x^{2} - 25) - (4 y^{2})$ .

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

Step 2.5.5.3.13

Simplify the expression.

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Step 2.5.5.3.13.1

Rewrite $- (4 x^{2} - 25 + 4 y^{2})$ as $- 1 (4 x^{2} - 25 + 4 y^{2})$ .

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

Step 2.5.5.3.13.2

Move the negative in front of the fraction.

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

Step 2.6

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

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

Step 3

Set the derivative equal to $0$ then solve the equation $- \frac{x (4 x^{2} - 25 + 4 y^{2})}{y (4 x^{2} + 4 y^{2} + 25)} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$x (4 x^{2} - 25 + 4 y^{2}) = 0$

Step 3.2

Solve the equation for $x$ .

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Step 3.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 x^{2} - 25 + 4 y^{2} = 0$

Step 3.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 3.2.3

Set $4 x^{2} - 25 + 4 y^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set $4 x^{2} - 25 + 4 y^{2}$ equal to $0$ .

$4 x^{2} - 25 + 4 y^{2} = 0$

Step 3.2.3.2

Solve $4 x^{2} - 25 + 4 y^{2} = 0$ for $x$ .

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Step 3.2.3.2.1

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

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Step 3.2.3.2.1.1

Add $25$ to both sides of the equation.

$4 x^{2} + 4 y^{2} = 25$

Step 3.2.3.2.1.2

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

$4 x^{2} = 25 - 4 y^{2}$

Step 3.2.3.2.2

Divide each term in $4 x^{2} = 25 - 4 y^{2}$ by $4$ and simplify.

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Step 3.2.3.2.2.1

Divide each term in $4 x^{2} = 25 - 4 y^{2}$ by $4$ .

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

Step 3.2.3.2.2.2

Simplify the left side.

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Step 3.2.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.3.2.2.2.1.2

Divide $x^{2}$ by $1$ .

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

Step 3.2.3.2.2.3

Simplify the right side.

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Step 3.2.3.2.2.3.1

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

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Step 3.2.3.2.2.3.1.1

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

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

Step 3.2.3.2.2.3.1.2

Cancel the common factors.

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Step 3.2.3.2.2.3.1.2.1

Factor $4$ out of $4$ .

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

Step 3.2.3.2.2.3.1.2.2

Cancel the common factor.

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

Step 3.2.3.2.2.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{25}{4} + \frac{- y^{2}}{1}$

Step 3.2.3.2.2.3.1.2.4

Divide $- y^{2}$ by $1$ .

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

Step 3.2.3.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{25}{4} - y^{2}}$

Step 3.2.3.2.4

Simplify $\pm \sqrt{\frac{25}{4} - y^{2}}$ .

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Step 3.2.3.2.4.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{\frac{5^{2}}{4} - y^{2}}$

Step 3.2.3.2.4.2

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{\frac{5^{2}}{2^{2}} - y^{2}}$

Step 3.2.3.2.4.3

Rewrite $\frac{5^{2}}{2^{2}}$ as ${(\frac{5}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{5}{2})}^{2} - y^{2}}$

Step 3.2.3.2.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{5}{2}$ and $b = y$ .

$x = \pm \sqrt{(\frac{5}{2} + y) (\frac{5}{2} - y)}$

Step 3.2.3.2.4.5

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

$x = \pm \sqrt{(\frac{5}{2} + y \cdot \frac{2}{2}) (\frac{5}{2} - y)}$

Step 3.2.3.2.4.6

Simplify terms.

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Step 3.2.3.2.4.6.1

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

$x = \pm \sqrt{(\frac{5}{2} + \frac{y \cdot 2}{2}) (\frac{5}{2} - y)}$

Step 3.2.3.2.4.6.2

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{5 + y \cdot 2}{2} (\frac{5}{2} - y)}$

Step 3.2.3.2.4.7

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

$x = \pm \sqrt{\frac{5 + 2 y}{2} (\frac{5}{2} - y)}$

Step 3.2.3.2.4.8

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

$x = \pm \sqrt{\frac{5 + 2 y}{2} (\frac{5}{2} - y \cdot \frac{2}{2})}$

Step 3.2.3.2.4.9

Combine $- y$ and $\frac{2}{2}$ .

$x = \pm \sqrt{\frac{5 + 2 y}{2} (\frac{5}{2} + \frac{- y \cdot 2}{2})}$

Step 3.2.3.2.4.10

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{5 + 2 y}{2} \cdot \frac{5 - y \cdot 2}{2}}$

Step 3.2.3.2.4.11

Multiply $2$ by $- 1$ .

$x = \pm \sqrt{\frac{5 + 2 y}{2} \cdot \frac{5 - 2 y}{2}}$

Step 3.2.3.2.4.12

Multiply $\frac{5 + 2 y}{2}$ by $\frac{5 - 2 y}{2}$ .

$x = \pm \sqrt{\frac{(5 + 2 y) (5 - 2 y)}{2 \cdot 2}}$

Step 3.2.3.2.4.13

Multiply $2$ by $2$ .

$x = \pm \sqrt{\frac{(5 + 2 y) (5 - 2 y)}{4}}$

Step 3.2.3.2.4.14

Rewrite $\frac{(5 + 2 y) (5 - 2 y)}{4}$ as ${(\frac{1}{2})}^{2} ((5 + 2 y) (5 - 2 y))$ .

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Step 3.2.3.2.4.14.1

Factor the perfect power $1^{2}$ out of $(5 + 2 y) (5 - 2 y)$ .

$x = \pm \sqrt{\frac{1^{2} ((5 + 2 y) (5 - 2 y))}{4}}$

Step 3.2.3.2.4.14.2

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{\frac{1^{2} ((5 + 2 y) (5 - 2 y))}{2^{2} \cdot 1}}$

Step 3.2.3.2.4.14.3

Rearrange the fraction $\frac{1^{2} ((5 + 2 y) (5 - 2 y))}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{{(\frac{1}{2})}^{2} ((5 + 2 y) (5 - 2 y))}$

Step 3.2.3.2.4.15

Pull terms out from under the radical.

$x = \pm \frac{1}{2} \sqrt{(5 + 2 y) (5 - 2 y)}$

Step 3.2.3.2.4.16

Combine $\frac{1}{2}$ and $\sqrt{(5 + 2 y) (5 - 2 y)}$ .

$x = \pm \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

Step 3.2.3.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.3.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

Step 3.2.3.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

Step 3.2.3.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

Step 3.2.4

The final solution is all the values that make $x (4 x^{2} - 25 + 4 y^{2}) = 0$ true.

$x = 0$

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = 0$

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = 0$

$x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

$x = - \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$

Step 4

Solve the function $f (x) = 25 (x^{2} - y^{2})$ at $x = 0$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 25 ({(0)}^{2} - y^{2})$

Step 4.2

Simplify the result.

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Step 4.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 25 (0 - y^{2})$

Step 4.2.2

Subtract $y^{2}$ from $0$ .

$f (0) = 25 (- y^{2})$

Step 4.2.3

Multiply $- 1$ by $25$ .

$f (0) = - 25 y^{2}$

Step 4.2.4

The final answer is $- 25 y^{2}$ .

$- 25 y^{2}$

Step 5

Solve the function $f (x) = 25 (x^{2} - y^{2})$ at $x = \frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$ .

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Step 5.1

Replace the variable $x$ with $\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$ in the expression.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 ({(\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2})}^{2} - y^{2})$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Apply the product rule to $\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{{\sqrt{(5 + 2 y) (5 - 2 y)}}^{2}}{2^{2}} - y^{2})$

Step 5.2.1.2

Simplify the numerator.

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Step 5.2.1.2.1

Rewrite ${\sqrt{(5 + 2 y) (5 - 2 y)}}^{2}$ as $(5 + 2 y) (5 - 2 y)$ .

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Step 5.2.1.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(5 + 2 y) (5 - 2 y)}$ as ${((5 + 2 y) (5 - 2 y))}^{\frac{1}{2}}$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{{({((5 + 2 y) (5 - 2 y))}^{\frac{1}{2}})}^{2}}{2^{2}} - y^{2})$

Step 5.2.1.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{{((5 + 2 y) (5 - 2 y))}^{\frac{1}{2} \cdot 2}}{2^{2}} - y^{2})$

Step 5.2.1.2.1.3

Combine $\frac{1}{2}$ and $2$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{{((5 + 2 y) (5 - 2 y))}^{\frac{2}{2}}}{2^{2}} - y^{2})$

Step 5.2.1.2.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.2.1.4.1

Cancel the common factor.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{{((5 + 2 y) (5 - 2 y))}^{\frac{2}{2}}}{2^{2}} - y^{2})$

Step 5.2.1.2.1.4.2

Rewrite the expression.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.1.5

Simplify.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.2

Expand $(5 + 2 y) (5 - 2 y)$ using the FOIL Method.

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Step 5.2.1.2.2.1

Apply the distributive property.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 (5 - 2 y) + 2 y (5 - 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.2.2

Apply the distributive property.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 5 (- 2 y) + 2 y (5 - 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.2.3

Apply the distributive property.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 5 (- 2 y) + 2 y \cdot 5 + 2 y (- 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.3

Combine the opposite terms in $5 \cdot 5 + 5 (- 2 y) + 2 y \cdot 5 + 2 y (- 2 y)$ .

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Step 5.2.1.2.3.1

Reorder the factors in the terms $5 (- 2 y)$ and $2 y \cdot 5$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 - 2 \cdot (5 y) + 2 \cdot (5 y) + 2 y (- 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.3.2

Add $- 2 \cdot 5 y$ and $2 \cdot 5 y$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 0 + 2 y (- 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.3.3

Add $5 \cdot 5$ and $0$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 2 y (- 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.4

Simplify each term.

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Step 5.2.1.2.4.1

Multiply $5$ by $5$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 y (- 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.2.4.2

Rewrite using the commutative property of multiplication.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 \cdot (- 2 y \cdot y)}{2^{2}} - y^{2})$

Step 5.2.1.2.4.3

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

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Step 5.2.1.2.4.3.1

Move $y$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 \cdot (- 2 (y \cdot y))}{2^{2}} - y^{2})$

Step 5.2.1.2.4.3.2

Multiply $y$ by $y$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 \cdot (- 2 y^{2})}{2^{2}} - y^{2})$

Step 5.2.1.2.4.4

Multiply $2$ by $- 2$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 - 4 y^{2}}{2^{2}} - y^{2})$

Step 5.2.1.2.5

Rewrite $25$ as $5^{2}$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5^{2} - 4 y^{2}}{2^{2}} - y^{2})$

Step 5.2.1.2.6

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5^{2} - {(2 y)}^{2}}{2^{2}} - y^{2})$

Step 5.2.1.2.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = 2 y$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - (2 y))}{2^{2}} - y^{2})$

Step 5.2.1.2.8

Multiply $2$ by $- 1$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - 2 y)}{2^{2}} - y^{2})$

Step 5.2.1.3

Raise $2$ to the power of $2$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - 2 y)}{4} - y^{2})$

Step 5.2.2

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

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - 2 y)}{4} - y^{2} \cdot \frac{4}{4})$

Step 5.2.3

Simplify terms.

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Step 5.2.3.1

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

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - 2 y)}{4} + \frac{- y^{2} \cdot 4}{4})$

Step 5.2.3.2

Combine the numerators over the common denominator.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{(5 + 2 y) (5 - 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Expand $(5 + 2 y) (5 - 2 y)$ using the FOIL Method.

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Step 5.2.4.1.1

Apply the distributive property.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 (5 - 2 y) + 2 y (5 - 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4.1.2

Apply the distributive property.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 5 (- 2 y) + 2 y (5 - 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4.1.3

Apply the distributive property.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 5 (- 2 y) + 2 y \cdot 5 + 2 y (- 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4.2

Combine the opposite terms in $5 \cdot 5 + 5 (- 2 y) + 2 y \cdot 5 + 2 y (- 2 y)$ .

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Step 5.2.4.2.1

Reorder the factors in the terms $5 (- 2 y)$ and $2 y \cdot 5$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 - 2 \cdot (5 y) + 2 \cdot (5 y) + 2 y (- 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4.2.2

Add $- 2 \cdot 5 y$ and $2 \cdot 5 y$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 0 + 2 y (- 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4.2.3

Add $5 \cdot 5$ and $0$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{5 \cdot 5 + 2 y (- 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4.3

Simplify each term.

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Step 5.2.4.3.1

Multiply $5$ by $5$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 y (- 2 y) - y^{2} \cdot 4}{4})$

Step 5.2.4.3.2

Rewrite using the commutative property of multiplication.

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 \cdot (- 2 y \cdot y) - y^{2} \cdot 4}{4})$

Step 5.2.4.3.3

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

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Step 5.2.4.3.3.1

Move $y$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 \cdot (- 2 (y \cdot y)) - y^{2} \cdot 4}{4})$

Step 5.2.4.3.3.2

Multiply $y$ by $y$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 + 2 \cdot (- 2 y^{2}) - y^{2} \cdot 4}{4})$

Step 5.2.4.3.4

Multiply $2$ by $- 2$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 - 4 y^{2} - y^{2} \cdot 4}{4})$

Step 5.2.4.4

Multiply $4$ by $- 1$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 - 4 y^{2} - 4 y^{2}}{4})$

Step 5.2.4.5

Subtract $4 y^{2}$ from $- 4 y^{2}$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = 25 (\frac{25 - 8 y^{2}}{4})$

Step 5.2.5

Combine $25$ and $\frac{25 - 8 y^{2}}{4}$ .

$f (\frac{\sqrt{(5 + 2 y) (5 - 2 y)}}{2}) = \frac{25 (25 - 8 y^{2})}{4}$

Step 5.2.6

The final answer is $\frac{25 (25 - 8 y^{2})}{4}$ .

$\frac{25 (25 - 8 y^{2})}{4}$

Step 6

The horizontal tangent lines are $y = - 25 y^{2}, y = \frac{25 (25 - 8 y^{2})}{4}, y = \frac{25 (25 - 8 y^{2})}{4}$

$y = - 25 y^{2}, y = \frac{25 (25 - 8 y^{2})}{4}, y = \frac{25 (25 - 8 y^{2})}{4}$

Step 7