Calculus Examples

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

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

Subtract $6$ from both sides of the equation.

$x y^{2} + x^{2} y - 6 = 0$

Step 1.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.3

Substitute the values $a = x$ , $b = x^{2}$ , and $c = - 6$ into the quadratic formula and solve for $y$ .

$\frac{- x^{2} \pm \sqrt{{(x^{2})}^{2} - 4 \cdot (x \cdot - 6)}}{2 x}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.4.1.1

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

$y = \frac{- x^{2} \pm \sqrt{x^{2 \cdot 2} - 4 \cdot x \cdot - 6}}{2 x}$

Step 1.4.1.2

Multiply $2$ by $2$ .

$y = \frac{- x^{2} \pm \sqrt{x^{4} - 4 \cdot x \cdot - 6}}{2 x}$

Step 1.4.2

Multiply $- 6$ by $- 4$ .

$y = \frac{- x^{2} \pm \sqrt{x^{4} + 24 x}}{2 x}$

Step 1.4.3

Factor $x$ out of $x^{4} + 24 x$ .

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Step 1.4.3.1

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

$y = \frac{- x^{2} \pm \sqrt{x \cdot x^{3} + 24 x}}{2 x}$

Step 1.4.3.2

Factor $x$ out of $24 x$ .

$y = \frac{- x^{2} \pm \sqrt{x \cdot x^{3} + x \cdot 24}}{2 x}$

Step 1.4.3.3

Factor $x$ out of $x \cdot x^{3} + x \cdot 24$ .

$y = \frac{- x^{2} \pm \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.5.1

Change the $\pm$ to $+$ .

$y = \frac{- x^{2} + \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.5.2

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

$y = \frac{- (x^{2}) + \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.5.3

Factor $- 1$ out of $\sqrt{x (x^{3} + 24)}$ .

$y = \frac{- (x^{2}) - 1 (- \sqrt{x (x^{3} + 24)})}{2 x}$

Step 1.5.4

Factor $- 1$ out of $- (x^{2}) - 1 (- \sqrt{x (x^{3} + 24)})$ .

$y = \frac{- (x^{2} - \sqrt{x (x^{3} + 24)})}{2 x}$

Step 1.5.5

Rewrite $- (x^{2} - \sqrt{x (x^{3} + 24)})$ as $- 1 (x^{2} - \sqrt{x (x^{3} + 24)})$ .

$y = \frac{- 1 (x^{2} - \sqrt{x (x^{3} + 24)})}{2 x}$

Step 1.5.6

Move the negative in front of the fraction.

$y = - \frac{x^{2} - \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 1.6.1.1.1

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

$y = \frac{- x^{2} \pm \sqrt{x^{2 \cdot 2} - 4 \cdot x \cdot - 6}}{2 x}$

Step 1.6.1.1.2

Multiply $2$ by $2$ .

$y = \frac{- x^{2} \pm \sqrt{x^{4} - 4 \cdot x \cdot - 6}}{2 x}$

Step 1.6.1.2

Multiply $- 6$ by $- 4$ .

$y = \frac{- x^{2} \pm \sqrt{x^{4} + 24 x}}{2 x}$

Step 1.6.1.3

Factor $x$ out of $x^{4} + 24 x$ .

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Step 1.6.1.3.1

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

$y = \frac{- x^{2} \pm \sqrt{x \cdot x^{3} + 24 x}}{2 x}$

Step 1.6.1.3.2

Factor $x$ out of $24 x$ .

$y = \frac{- x^{2} \pm \sqrt{x \cdot x^{3} + x \cdot 24}}{2 x}$

Step 1.6.1.3.3

Factor $x$ out of $x \cdot x^{3} + x \cdot 24$ .

$y = \frac{- x^{2} \pm \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.6.2

Change the $\pm$ to $-$ .

$y = \frac{- x^{2} - \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.6.3

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

$y = \frac{- (x^{2}) - \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.6.4

Factor $- 1$ out of $- \sqrt{x (x^{3} + 24)}$ .

$y = \frac{- (x^{2}) - (\sqrt{x (x^{3} + 24)})}{2 x}$

Step 1.6.5

Factor $- 1$ out of $- (x^{2}) - (\sqrt{x (x^{3} + 24)})$ .

$y = \frac{- (x^{2} + \sqrt{x (x^{3} + 24)})}{2 x}$

Step 1.6.6

Rewrite $- (x^{2} + \sqrt{x (x^{3} + 24)})$ as $- 1 (x^{2} + \sqrt{x (x^{3} + 24)})$ .

$y = \frac{- 1 (x^{2} + \sqrt{x (x^{3} + 24)})}{2 x}$

Step 1.6.7

Move the negative in front of the fraction.

$y = - \frac{x^{2} + \sqrt{x (x^{3} + 24)}}{2 x}$

Step 1.7

The final answer is the combination of both solutions.

$y = - \frac{x^{2} - \sqrt{x (x^{3} + 24)}}{2 x}$

$y = - \frac{x^{2} + \sqrt{x (x^{3} + 24)}}{2 x}$

$y = - \frac{x^{2} - \sqrt{x (x^{3} + 24)}}{2 x}$

$y = - \frac{x^{2} + \sqrt{x (x^{3} + 24)}}{2 x}$

Step 2

Set each solution of $y$ as a function of $x$ .

$y = - \frac{x^{2} - \sqrt{x (x^{3} + 24)}}{2 x} \to f (x) = - \frac{x^{2} - \sqrt{x (x^{3} + 24)}}{2 x}$

$y = - \frac{x^{2} + \sqrt{x (x^{3} + 24)}}{2 x} \to f (x) = - \frac{x^{2} + \sqrt{x (x^{3} + 24)}}{2 x}$

Step 3

Because the $y$ variable in the equation $x y^{2} + x^{2} y = 6$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

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

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

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

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

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

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 $g (x) = y$ .

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Step 3.2.2.2.1

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

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

Step 3.2.2.2.2

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

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

Step 3.2.2.2.3

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

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

Step 3.2.2.3

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

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

Step 3.2.2.4

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

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

Step 3.2.2.5

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

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

Step 3.2.2.6

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

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

Step 3.2.3

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

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Step 3.2.3.1

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 $g (x) = y$ .

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.3.4

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

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

Step 3.2.4

Reorder terms.

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

Step 3.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0$

Step 3.4

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

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

Step 3.5

Solve for $y'$ .

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Step 3.5.1

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

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Step 3.5.1.1

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

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

Step 3.5.1.2

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

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

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

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

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

Step 3.5.2.3

Factor $x y'$ out of $x y' (2 y) + x y' x$ .

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

Step 3.5.3

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

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Step 3.5.3.1

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

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

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

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

Step 3.5.3.2.1.2

Rewrite the expression.

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

Step 3.5.3.2.2

Cancel the common factor of $2 y + x$ .

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Step 3.5.3.2.2.1

Cancel the common factor.

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

Step 3.5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Simplify each term.

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Step 3.5.3.3.1.1

Move the negative in front of the fraction.

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

Step 3.5.3.3.1.2

Cancel the common factor of $x$ .

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Step 3.5.3.3.1.2.1

Cancel the common factor.

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

Step 3.5.3.3.1.2.2

Rewrite the expression.

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

Step 3.5.3.3.1.3

Move the negative in front of the fraction.

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

Step 3.5.3.3.2

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

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

Step 3.5.3.3.3

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

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Step 3.5.3.3.3.1

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

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

Step 3.5.3.3.3.2

Reorder the factors of $(2 y + x) x$ .

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

Step 3.5.3.3.4

Combine the numerators over the common denominator.

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

Step 3.5.3.3.5

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

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Step 3.5.3.3.5.1

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

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

Step 3.5.3.3.5.2

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

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

Step 3.5.3.3.5.3

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

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

Step 3.5.3.3.6

Factor $- 1$ out of $- y$ .

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

Step 3.5.3.3.7

Factor $- 1$ out of $- 2 x$ .

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

Step 3.5.3.3.8

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

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

Step 3.5.3.3.9

Simplify the expression.

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Step 3.5.3.3.9.1

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

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

Step 3.5.3.3.9.2

Move the negative in front of the fraction.

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

Step 3.6

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

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

Step 4

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

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Step 4.1

Set the numerator equal to zero.

$y (y + 2 x) = 0$

Step 4.2

Solve the equation for $x$ .

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Step 4.2.1

Divide each term in $y (y + 2 x) = 0$ by $y$ and simplify.

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Step 4.2.1.1

Divide each term in $y (y + 2 x) = 0$ by $y$ .

$\frac{y (y + 2 x)}{y} = \frac{0}{y}$

Step 4.2.1.2

Simplify the left side.

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Step 4.2.1.2.1

Cancel the common factor of $y$ .

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Step 4.2.1.2.1.1

Cancel the common factor.

$\frac{y (y + 2 x)}{y} = \frac{0}{y}$

Step 4.2.1.2.1.2

Divide $y + 2 x$ by $1$ .

$y + 2 x = \frac{0}{y}$

Step 4.2.1.3

Simplify the right side.

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Step 4.2.1.3.1

Divide $0$ by $y$ .

$y + 2 x = 0$

Step 4.2.2

Subtract $y$ from both sides of the equation.

$2 x = - y$

Step 4.2.3

Divide each term in $2 x = - y$ by $2$ and simplify.

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Step 4.2.3.1

Divide each term in $2 x = - y$ by $2$ .

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

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Cancel the common factor of $2$ .

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Step 4.2.3.2.1.1

Cancel the common factor.

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

Step 4.2.3.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Move the negative in front of the fraction.

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

Step 5

Solve the function $f (x) = - \frac{x^{2} - \sqrt{x (x^{3} + 24)}}{2 x}$ at $x = - \frac{y}{2}$ .

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Step 5.1

Replace the variable $x$ with $- \frac{y}{2}$ in the expression.

$f (- \frac{y}{2}) = - \frac{{(- \frac{y}{2})}^{2} - \sqrt{(- \frac{y}{2}) ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1

Apply the product rule to $- \frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{{(- 1)}^{2} {(\frac{y}{2})}^{2} - \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.1.2

Apply the product rule to $\frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{{(- 1)}^{2} (\frac{y^{2}}{2^{2}}) - \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{y}{2}) = - \frac{1 (\frac{y^{2}}{2^{2}}) - \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.3

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{2^{2}} - \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.4

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.5.1

Apply the product rule to $- \frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot ({(- 1)}^{3} {(\frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.5.2

Apply the product rule to $\frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot ({(- 1)}^{3} (\frac{y^{3}}{2^{3}}) + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.6

Raise $- 1$ to the power of $3$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{2^{3}} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.7

Raise $2$ to the power of $3$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{8} + 24)}}{2 (- \frac{y}{2})}$

Step 5.2.1.8

To write $24$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{8} + 24 \cdot \frac{8}{8})}}{2 (- \frac{y}{2})}$

Step 5.2.1.9

Combine $24$ and $\frac{8}{8}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{8} + \frac{24 \cdot 8}{8})}}{2 (- \frac{y}{2})}$

Step 5.2.1.10

Combine the numerators over the common denominator.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot \frac{- y^{3} + 24 \cdot 8}{8}}}{2 (- \frac{y}{2})}$

Step 5.2.1.11

Multiply $24$ by $8$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{y}{2} \cdot \frac{- y^{3} + 192}{8}}}{2 (- \frac{y}{2})}$

Step 5.2.1.12

Combine exponents.

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Step 5.2.1.12.1

Multiply $\frac{- y^{3} + 192}{8}$ by $\frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{(- y^{3} + 192) y}{8 \cdot 2}}}{2 (- \frac{y}{2})}$

Step 5.2.1.12.2

Multiply $8$ by $2$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{(- y^{3} + 192) y}{16}}}{2 (- \frac{y}{2})}$

Step 5.2.1.13

Rewrite $- \frac{(- y^{3} + 192) y}{16}$ as ${({(\frac{1}{2})}^{2})}^{2} \cdot (- (- y^{3} + 192) y)$ .

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Step 5.2.1.13.1

Factor the perfect power $1^{2}$ out of $(- y^{3} + 192) y$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{1^{2} ((- y^{3} + 192) y)}{16}}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.2

Factor the perfect power $4^{2}$ out of $16$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- \frac{1^{2} ((- y^{3} + 192) y)}{4^{2} \cdot 1}}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.3

Rearrange the fraction $\frac{1^{2} ((- y^{3} + 192) y)}{4^{2} \cdot 1}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{- ({(\frac{1}{4})}^{2} ((- y^{3} + 192) y))}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.4

Reorder $- 1$ and ${(\frac{1}{4})}^{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{{(\frac{1}{4})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.5

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{{(\frac{1^{2}}{4})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.6

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{{(\frac{1^{2}}{2^{2}})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.7

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{{({(\frac{1}{2})}^{2})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.8

Add parentheses.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{{({(\frac{1}{2})}^{2})}^{2} \cdot (- 1 ((- y^{3} + 192) y))}}{2 (- \frac{y}{2})}$

Step 5.2.1.13.9

Add parentheses.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \sqrt{{({(\frac{1}{2})}^{2})}^{2} \cdot (- (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 5.2.1.14

Pull terms out from under the radical.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - ({(\frac{1}{2})}^{2} \sqrt{- (- y^{3} + 192) y})}{2 (- \frac{y}{2})}$

Step 5.2.1.15

Apply the product rule to $\frac{1}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - (\frac{1^{2}}{2^{2}} \cdot \sqrt{- (- y^{3} + 192) y})}{2 (- \frac{y}{2})}$

Step 5.2.1.16

One to any power is one.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - (\frac{1}{2^{2}} \cdot \sqrt{- (- y^{3} + 192) y})}{2 (- \frac{y}{2})}$

Step 5.2.1.17

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - (\frac{1}{4} \cdot \sqrt{- (- y^{3} + 192) y})}{2 (- \frac{y}{2})}$

Step 5.2.1.18

Combine $\frac{1}{4}$ and $\sqrt{- (- y^{3} + 192) y}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} - \frac{\sqrt{- (- y^{3} + 192) y}}{4}}{2 (- \frac{y}{2})}$

Step 5.2.1.19

Combine the numerators over the common denominator.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{2 (- \frac{y}{2})}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Multiply $- 1$ by $2$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{- 2 \frac{y}{2}}$

Step 5.2.2.2

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{- 2 y}{2}}$

Step 5.2.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.1

Reduce the expression $\frac{- 2 y}{2}$ by cancelling the common factors.

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Step 5.2.3.1.1

Factor $2$ out of $- 2 y$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{2 (- y)}{2}}$

Step 5.2.3.1.2

Factor $2$ out of $2$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{2 (- y)}{2 (1)}}$

Step 5.2.3.1.3

Cancel the common factor.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{2 (- y)}{2 \cdot 1}}$

Step 5.2.3.1.4

Rewrite the expression.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{- y}{1}}$

Step 5.2.3.2

Divide $- y$ by $1$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}}{- y}$

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{y}{2}) = - (\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4} \cdot \frac{1}{- y})$

Step 5.2.5

Cancel the common factor of $1$ and $- 1$ .

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Step 5.2.5.1

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

$f (- \frac{y}{2}) = - (\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4} \cdot \frac{- 1 \cdot - 1}{- y})$

Step 5.2.5.2

Move the negative in front of the fraction.

$f (- \frac{y}{2}) = - (\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4} \cdot (- \frac{1}{y}))$

Step 5.2.6

Rewrite using the commutative property of multiplication.

$f (- \frac{y}{2}) = - (- \frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4} \cdot \frac{1}{y})$

Step 5.2.7

Multiply $\frac{1}{y}$ by $\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4}$ .

$f (- \frac{y}{2}) = \frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{y \cdot 4}$

Step 5.2.8

Move $4$ to the left of $y$ .

$f (- \frac{y}{2}) = \frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4 \cdot y}$

Step 5.2.9

Multiply $- - \frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4 y}$ .

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Step 5.2.9.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{y}{2}) = 1 (\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4 y})$

Step 5.2.9.2

Multiply $\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4 y}$ by $1$ .

$f (- \frac{y}{2}) = \frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4 y}$

Step 5.2.10

Reorder factors in $\frac{y^{2} - \sqrt{- (- y^{3} + 192) y}}{4 y}$ .

$f (- \frac{y}{2}) = \frac{y^{2} - \sqrt{- y (- y^{3} + 192)}}{4 y}$

Step 5.2.11

The final answer is $\frac{y^{2} - \sqrt{- y (- y^{3} + 192)}}{4 y}$ .

$\frac{y^{2} - \sqrt{- y (- y^{3} + 192)}}{4 y}$

Step 6

Solve the function $f (x) = - \frac{x^{2} + \sqrt{x (x^{3} + 24)}}{2 x}$ at $x = - \frac{y}{2}$ .

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Step 6.1

Replace the variable $x$ with $- \frac{y}{2}$ in the expression.

$f (- \frac{y}{2}) = - \frac{{(- \frac{y}{2})}^{2} + \sqrt{(- \frac{y}{2}) ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.2.1.1.1

Apply the product rule to $- \frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{{(- 1)}^{2} {(\frac{y}{2})}^{2} + \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.1.2

Apply the product rule to $\frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{{(- 1)}^{2} (\frac{y^{2}}{2^{2}}) + \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{y}{2}) = - \frac{1 (\frac{y^{2}}{2^{2}}) + \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.3

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{2^{2}} + \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.4

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot ({(- \frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.2.1.5.1

Apply the product rule to $- \frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot ({(- 1)}^{3} {(\frac{y}{2})}^{3} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.5.2

Apply the product rule to $\frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot ({(- 1)}^{3} (\frac{y^{3}}{2^{3}}) + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.6

Raise $- 1$ to the power of $3$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{2^{3}} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.7

Raise $2$ to the power of $3$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{8} + 24)}}{2 (- \frac{y}{2})}$

Step 6.2.1.8

To write $24$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{8} + 24 \cdot \frac{8}{8})}}{2 (- \frac{y}{2})}$

Step 6.2.1.9

Combine $24$ and $\frac{8}{8}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot (- \frac{y^{3}}{8} + \frac{24 \cdot 8}{8})}}{2 (- \frac{y}{2})}$

Step 6.2.1.10

Combine the numerators over the common denominator.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot \frac{- y^{3} + 24 \cdot 8}{8}}}{2 (- \frac{y}{2})}$

Step 6.2.1.11

Multiply $24$ by $8$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{y}{2} \cdot \frac{- y^{3} + 192}{8}}}{2 (- \frac{y}{2})}$

Step 6.2.1.12

Combine exponents.

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Step 6.2.1.12.1

Multiply $\frac{- y^{3} + 192}{8}$ by $\frac{y}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{(- y^{3} + 192) y}{8 \cdot 2}}}{2 (- \frac{y}{2})}$

Step 6.2.1.12.2

Multiply $8$ by $2$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{(- y^{3} + 192) y}{16}}}{2 (- \frac{y}{2})}$

Step 6.2.1.13

Rewrite $- \frac{(- y^{3} + 192) y}{16}$ as ${({(\frac{1}{2})}^{2})}^{2} \cdot (- (- y^{3} + 192) y)$ .

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Step 6.2.1.13.1

Factor the perfect power $1^{2}$ out of $(- y^{3} + 192) y$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{1^{2} ((- y^{3} + 192) y)}{16}}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.2

Factor the perfect power $4^{2}$ out of $16$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- \frac{1^{2} ((- y^{3} + 192) y)}{4^{2} \cdot 1}}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.3

Rearrange the fraction $\frac{1^{2} ((- y^{3} + 192) y)}{4^{2} \cdot 1}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{- ({(\frac{1}{4})}^{2} ((- y^{3} + 192) y))}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.4

Reorder $- 1$ and ${(\frac{1}{4})}^{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{{(\frac{1}{4})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.5

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{{(\frac{1^{2}}{4})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.6

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{{(\frac{1^{2}}{2^{2}})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.7

Rewrite $\frac{1^{2}}{2^{2}}$ as ${(\frac{1}{2})}^{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{{({(\frac{1}{2})}^{2})}^{2} \cdot (- 1 (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.8

Add parentheses.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{{({(\frac{1}{2})}^{2})}^{2} \cdot (- 1 ((- y^{3} + 192) y))}}{2 (- \frac{y}{2})}$

Step 6.2.1.13.9

Add parentheses.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \sqrt{{({(\frac{1}{2})}^{2})}^{2} \cdot (- (- y^{3} + 192) y)}}{2 (- \frac{y}{2})}$

Step 6.2.1.14

Pull terms out from under the radical.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + {(\frac{1}{2})}^{2} \sqrt{- (- y^{3} + 192) y}}{2 (- \frac{y}{2})}$

Step 6.2.1.15

Apply the product rule to $\frac{1}{2}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \frac{1^{2}}{2^{2}} \cdot \sqrt{- (- y^{3} + 192) y}}{2 (- \frac{y}{2})}$

Step 6.2.1.16

One to any power is one.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \frac{1}{2^{2}} \cdot \sqrt{- (- y^{3} + 192) y}}{2 (- \frac{y}{2})}$

Step 6.2.1.17

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \frac{1}{4} \cdot \sqrt{- (- y^{3} + 192) y}}{2 (- \frac{y}{2})}$

Step 6.2.1.18

Combine $\frac{1}{4}$ and $\sqrt{- (- y^{3} + 192) y}$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2}}{4} + \frac{\sqrt{- (- y^{3} + 192) y}}{4}}{2 (- \frac{y}{2})}$

Step 6.2.1.19

Combine the numerators over the common denominator.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{2 (- \frac{y}{2})}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Multiply $- 1$ by $2$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{- 2 \frac{y}{2}}$

Step 6.2.2.2

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

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{- 2 y}{2}}$

Step 6.2.3

Reduce the expression by cancelling the common factors.

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Step 6.2.3.1

Reduce the expression $\frac{- 2 y}{2}$ by cancelling the common factors.

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Step 6.2.3.1.1

Factor $2$ out of $- 2 y$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{2 (- y)}{2}}$

Step 6.2.3.1.2

Factor $2$ out of $2$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{2 (- y)}{2 (1)}}$

Step 6.2.3.1.3

Cancel the common factor.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{2 (- y)}{2 \cdot 1}}$

Step 6.2.3.1.4

Rewrite the expression.

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{\frac{- y}{1}}$

Step 6.2.3.2

Divide $- y$ by $1$ .

$f (- \frac{y}{2}) = - \frac{\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}}{- y}$

Step 6.2.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{y}{2}) = - (\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4} \cdot \frac{1}{- y})$

Step 6.2.5

Cancel the common factor of $1$ and $- 1$ .

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Step 6.2.5.1

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

$f (- \frac{y}{2}) = - (\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4} \cdot \frac{- 1 \cdot - 1}{- y})$

Step 6.2.5.2

Move the negative in front of the fraction.

$f (- \frac{y}{2}) = - (\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4} \cdot (- \frac{1}{y}))$

Step 6.2.6

Rewrite using the commutative property of multiplication.

$f (- \frac{y}{2}) = - (- \frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4} \cdot \frac{1}{y})$

Step 6.2.7

Multiply $\frac{1}{y}$ by $\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4}$ .

$f (- \frac{y}{2}) = \frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{y \cdot 4}$

Step 6.2.8

Move $4$ to the left of $y$ .

$f (- \frac{y}{2}) = \frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4 \cdot y}$

Step 6.2.9

Multiply $- - \frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4 y}$ .

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Step 6.2.9.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{y}{2}) = 1 (\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4 y})$

Step 6.2.9.2

Multiply $\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4 y}$ by $1$ .

$f (- \frac{y}{2}) = \frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4 y}$

Step 6.2.10

Reorder factors in $\frac{y^{2} + \sqrt{- (- y^{3} + 192) y}}{4 y}$ .

$f (- \frac{y}{2}) = \frac{y^{2} + \sqrt{- y (- y^{3} + 192)}}{4 y}$

Step 6.2.11

The final answer is $\frac{y^{2} + \sqrt{- y (- y^{3} + 192)}}{4 y}$ .

$\frac{y^{2} + \sqrt{- y (- y^{3} + 192)}}{4 y}$

Step 7

The horizontal tangent lines are $y = \frac{y^{2} - \sqrt{- y (- y^{3} + 192)}}{4 y}, y = \frac{y^{2} + \sqrt{- y (- y^{3} + 192)}}{4 y}$

$y = \frac{y^{2} - \sqrt{- y (- y^{3} + 192)}}{4 y}, y = \frac{y^{2} + \sqrt{- y (- y^{3} + 192)}}{4 y}$

Step 8