Calculus Examples

$x^{2} - x y + 2 y^{2} = 1$

Step 1

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

Tap for more steps...

Step 1.1

Subtract $1$ from both sides of the equation.

$x^{2} - x y + 2 y^{2} - 1 = 0$

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 $a = 2$ , $b = - x$ , and $c = x^{2} - 1$ into the quadratic formula and solve for $y$ .

$\frac{x \pm \sqrt{{(- x)}^{2} - 4 \cdot (2 \cdot (x^{2} - 1))}}{2 \cdot 2}$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Simplify the numerator.

Tap for more steps...

Step 1.4.1.1

Apply the product rule to $- x$ .

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

Step 1.4.1.2

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

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

Step 1.4.1.3

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

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

Step 1.4.1.4

Multiply $- 4$ by $2$ .

$y = \frac{x \pm \sqrt{x^{2} - 8 \cdot (x^{2} - 1)}}{2 \cdot 2}$

Step 1.4.1.5

Apply the distributive property.

$y = \frac{x \pm \sqrt{x^{2} - 8 x^{2} - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.4.1.6

Multiply $- 8$ by $- 1$ .

$y = \frac{x \pm \sqrt{x^{2} - 8 x^{2} + 8}}{2 \cdot 2}$

Step 1.4.1.7

Subtract $8 x^{2}$ from $x^{2}$ .

$y = \frac{x \pm \sqrt{- 7 x^{2} + 8}}{2 \cdot 2}$

Step 1.4.2

Multiply $2$ by $2$ .

$y = \frac{x \pm \sqrt{- 7 x^{2} + 8}}{4}$

Step 1.5

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

Tap for more steps...

Step 1.5.1

Simplify the numerator.

Tap for more steps...

Step 1.5.1.1

Apply the product rule to $- x$ .

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

Step 1.5.1.2

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

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

Step 1.5.1.3

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

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

Step 1.5.1.4

Multiply $- 4$ by $2$ .

$y = \frac{x \pm \sqrt{x^{2} - 8 \cdot (x^{2} - 1)}}{2 \cdot 2}$

Step 1.5.1.5

Apply the distributive property.

$y = \frac{x \pm \sqrt{x^{2} - 8 x^{2} - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.5.1.6

Multiply $- 8$ by $- 1$ .

$y = \frac{x \pm \sqrt{x^{2} - 8 x^{2} + 8}}{2 \cdot 2}$

Step 1.5.1.7

Subtract $8 x^{2}$ from $x^{2}$ .

$y = \frac{x \pm \sqrt{- 7 x^{2} + 8}}{2 \cdot 2}$

Step 1.5.2

Multiply $2$ by $2$ .

$y = \frac{x \pm \sqrt{- 7 x^{2} + 8}}{4}$

Step 1.5.3

Change the $\pm$ to $+$ .

$y = \frac{x + \sqrt{- 7 x^{2} + 8}}{4}$

Step 1.6

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

Tap for more steps...

Step 1.6.1

Simplify the numerator.

Tap for more steps...

Step 1.6.1.1

Apply the product rule to $- x$ .

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

Step 1.6.1.2

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

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

Step 1.6.1.3

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

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

Step 1.6.1.4

Multiply $- 4$ by $2$ .

$y = \frac{x \pm \sqrt{x^{2} - 8 \cdot (x^{2} - 1)}}{2 \cdot 2}$

Step 1.6.1.5

Apply the distributive property.

$y = \frac{x \pm \sqrt{x^{2} - 8 x^{2} - 8 \cdot - 1}}{2 \cdot 2}$

Step 1.6.1.6

Multiply $- 8$ by $- 1$ .

$y = \frac{x \pm \sqrt{x^{2} - 8 x^{2} + 8}}{2 \cdot 2}$

Step 1.6.1.7

Subtract $8 x^{2}$ from $x^{2}$ .

$y = \frac{x \pm \sqrt{- 7 x^{2} + 8}}{2 \cdot 2}$

Step 1.6.2

Multiply $2$ by $2$ .

$y = \frac{x \pm \sqrt{- 7 x^{2} + 8}}{4}$

Step 1.6.3

Change the $\pm$ to $-$ .

$y = \frac{x - \sqrt{- 7 x^{2} + 8}}{4}$

Step 1.7

The final answer is the combination of both solutions.

$y = \frac{x + \sqrt{- 7 x^{2} + 8}}{4}$

$y = \frac{x - \sqrt{- 7 x^{2} + 8}}{4}$

$y = \frac{x + \sqrt{- 7 x^{2} + 8}}{4}$

$y = \frac{x - \sqrt{- 7 x^{2} + 8}}{4}$

Step 2

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

$y = \frac{x + \sqrt{- 7 x^{2} + 8}}{4} \to f (x) = \frac{x + \sqrt{- 7 x^{2} + 8}}{4}$

$y = \frac{x - \sqrt{- 7 x^{2} + 8}}{4} \to f (x) = \frac{x - \sqrt{- 7 x^{2} + 8}}{4}$

Step 3

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

Tap for more steps...

Step 3.1

Differentiate both sides of the equation.

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

Step 3.2

Differentiate the left side of the equation.

Tap for more steps...

Step 3.2.1

Differentiate.

Tap for more steps...

Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

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

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

Step 3.2.2.2

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 $g (x) = y$ .

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

Step 3.2.2.3

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

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

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 $n = 1$ .

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

Step 3.2.2.5

Multiply $y$ by $1$ .

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

Step 3.2.3

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

Tap for more steps...

Step 3.2.3.1

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

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

Step 3.2.3.2

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 $g (x) = y$ .

Tap for more steps...

Step 3.2.3.2.1

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

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

Step 3.2.3.2.2

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

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

Step 3.2.3.2.3

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

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

Step 3.2.3.3

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

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

Step 3.2.3.4

Multiply $2$ by $2$ .

$2 x - (x y' + y) + 4 y y'$

Step 3.2.4

Simplify.

Tap for more steps...

Step 3.2.4.1

Apply the distributive property.

$2 x - (x y') - y + 4 y y'$

Step 3.2.4.2

Remove unnecessary parentheses.

$2 x - x y' - y + 4 y y'$

Step 3.2.4.3

Reorder terms.

$- x y' + 4 y y' + 2 x - y$

Step 3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 3.4

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

$- x y' + 4 y y' + 2 x - y = 0$

Step 3.5

Solve for $y'$ .

Tap for more steps...

Step 3.5.1

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

Tap for more steps...

Step 3.5.1.1

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

$- x y' + 4 y y' - y = - 2 x$

Step 3.5.1.2

Add $y$ to both sides of the equation.

$- x y' + 4 y y' = - 2 x + y$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

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

$y' (- x) + 4 y y' = - 2 x + y$

Step 3.5.2.2

Factor $y'$ out of $4 y y'$ .

$y' (- x) + y' (4 y) = - 2 x + y$

Step 3.5.2.3

Factor $y'$ out of $y' (- x) + y' (4 y)$ .

$y' (- x + 4 y) = - 2 x + y$

Step 3.5.3

Divide each term in $y' (- x + 4 y) = - 2 x + y$ by $- x + 4 y$ and simplify.

Tap for more steps...

Step 3.5.3.1

Divide each term in $y' (- x + 4 y) = - 2 x + y$ by $- x + 4 y$ .

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

Step 3.5.3.2

Simplify the left side.

Tap for more steps...

Step 3.5.3.2.1

Cancel the common factor of $- x + 4 y$ .

Tap for more steps...

Step 3.5.3.2.1.1

Cancel the common factor.

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

Step 3.5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 3.5.3.3

Simplify the right side.

Tap for more steps...

Step 3.5.3.3.1

Combine the numerators over the common denominator.

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

Step 3.5.3.3.2

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

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

Step 3.5.3.3.3

Factor $- 1$ out of $y$ .

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

Step 3.5.3.3.4

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

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

Step 3.5.3.3.5

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

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

Step 3.5.3.3.6

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

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

Step 3.5.3.3.7

Factor $- 1$ out of $4 y$ .

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

Step 3.5.3.3.8

Factor $- 1$ out of $- (x) - (- 4 y)$ .

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

Step 3.5.3.3.9

Rewrite $- (x - 4 y)$ as $- 1 (x - 4 y)$ .

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

Step 3.5.3.3.10

Cancel the common factor.

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

Step 3.5.3.3.11

Rewrite the expression.

$y' = \frac{2 x - y}{x - 4 y}$

Step 3.6

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

$\frac{d y}{d x} = \frac{2 x - y}{x - 4 y}$

Step 4

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

Tap for more steps...

Step 4.1

Set the numerator equal to zero.

$2 x - y = 0$

Step 4.2

Solve the equation for $x$ .

Tap for more steps...

Step 4.2.1

Add $y$ to both sides of the equation.

$2 x = y$

Step 4.2.2

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

Tap for more steps...

Step 4.2.2.1

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

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

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 5

Solve the function $f (x) = \frac{x + \sqrt{- 7 x^{2} + 8}}{4}$ at $x = \frac{y}{2}$ .

Tap for more steps...

Step 5.1

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

$f (\frac{y}{2}) = \frac{(\frac{y}{2}) + \sqrt{- 7 {(\frac{y}{2})}^{2} + 8}}{4}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Simplify the numerator.

Tap for more steps...

Step 5.2.1.1

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

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{- 7 \frac{y^{2}}{2^{2}} + 8}}{4}$

Step 5.2.1.2

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

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{- 7 \frac{y^{2}}{4} + 8}}{4}$

Step 5.2.1.3

Combine $- 7$ and $\frac{y^{2}}{4}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{\frac{- 7 y^{2}}{4} + 8}}{4}$

Step 5.2.1.4

Move the negative in front of the fraction.

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{- \frac{7 y^{2}}{4} + 8}}{4}$

Step 5.2.1.5

To write $8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{- \frac{7 y^{2}}{4} + 8 \cdot \frac{4}{4}}}{4}$

Step 5.2.1.6

Combine $8$ and $\frac{4}{4}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{- \frac{7 y^{2}}{4} + \frac{8 \cdot 4}{4}}}{4}$

Step 5.2.1.7

Combine the numerators over the common denominator.

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{\frac{- 7 y^{2} + 8 \cdot 4}{4}}}{4}$

Step 5.2.1.8

Multiply $8$ by $4$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{\frac{- 7 y^{2} + 32}{4}}}{4}$

Step 5.2.1.9

Rewrite $\frac{- 7 y^{2} + 32}{4}$ as ${(\frac{1}{2})}^{2} (- 7 y^{2} + 32)$ .

Tap for more steps...

Step 5.2.1.9.1

Factor the perfect power $1^{2}$ out of $- 7 y^{2} + 32$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{\frac{1^{2} (- 7 y^{2} + 32)}{4}}}{4}$

Step 5.2.1.9.2

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

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{\frac{1^{2} (- 7 y^{2} + 32)}{2^{2} \cdot 1}}}{4}$

Step 5.2.1.9.3

Rearrange the fraction $\frac{1^{2} (- 7 y^{2} + 32)}{2^{2} \cdot 1}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \sqrt{{(\frac{1}{2})}^{2} (- 7 y^{2} + 32)}}{4}$

Step 5.2.1.10

Pull terms out from under the radical.

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \frac{1}{2} \cdot \sqrt{- 7 y^{2} + 32}}{4}$

Step 5.2.1.11

Combine $\frac{1}{2}$ and $\sqrt{- 7 y^{2} + 32}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} + \frac{\sqrt{- 7 y^{2} + 32}}{2}}{4}$

Step 5.2.1.12

Combine the numerators over the common denominator.

$f (\frac{y}{2}) = \frac{\frac{y + \sqrt{- 7 y^{2} + 32}}{2}}{4}$

Step 5.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{y}{2}) = \frac{y + \sqrt{- 7 y^{2} + 32}}{2} \cdot \frac{1}{4}$

Step 5.2.3

Multiply $\frac{y + \sqrt{- 7 y^{2} + 32}}{2} \cdot \frac{1}{4}$ .

Tap for more steps...

Step 5.2.3.1

Multiply $\frac{y + \sqrt{- 7 y^{2} + 32}}{2}$ by $\frac{1}{4}$ .

$f (\frac{y}{2}) = \frac{y + \sqrt{- 7 y^{2} + 32}}{2 \cdot 4}$

Step 5.2.3.2

Multiply $2$ by $4$ .

$f (\frac{y}{2}) = \frac{y + \sqrt{- 7 y^{2} + 32}}{8}$

Step 5.2.4

The final answer is $\frac{y + \sqrt{- 7 y^{2} + 32}}{8}$ .

$\frac{y + \sqrt{- 7 y^{2} + 32}}{8}$

Step 6

Solve the function $f (x) = \frac{x - \sqrt{- 7 x^{2} + 8}}{4}$ at $x = \frac{y}{2}$ .

Tap for more steps...

Step 6.1

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

$f (\frac{y}{2}) = \frac{(\frac{y}{2}) - \sqrt{- 7 {(\frac{y}{2})}^{2} + 8}}{4}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Simplify the numerator.

Tap for more steps...

Step 6.2.1.1

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

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{- 7 \frac{y^{2}}{2^{2}} + 8}}{4}$

Step 6.2.1.2

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

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{- 7 \frac{y^{2}}{4} + 8}}{4}$

Step 6.2.1.3

Combine $- 7$ and $\frac{y^{2}}{4}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{\frac{- 7 y^{2}}{4} + 8}}{4}$

Step 6.2.1.4

Move the negative in front of the fraction.

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{- \frac{7 y^{2}}{4} + 8}}{4}$

Step 6.2.1.5

To write $8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{- \frac{7 y^{2}}{4} + 8 \cdot \frac{4}{4}}}{4}$

Step 6.2.1.6

Combine $8$ and $\frac{4}{4}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{- \frac{7 y^{2}}{4} + \frac{8 \cdot 4}{4}}}{4}$

Step 6.2.1.7

Combine the numerators over the common denominator.

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{\frac{- 7 y^{2} + 8 \cdot 4}{4}}}{4}$

Step 6.2.1.8

Multiply $8$ by $4$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{\frac{- 7 y^{2} + 32}{4}}}{4}$

Step 6.2.1.9

Rewrite $\frac{- 7 y^{2} + 32}{4}$ as ${(\frac{1}{2})}^{2} (- 7 y^{2} + 32)$ .

Tap for more steps...

Step 6.2.1.9.1

Factor the perfect power $1^{2}$ out of $- 7 y^{2} + 32$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{\frac{1^{2} (- 7 y^{2} + 32)}{4}}}{4}$

Step 6.2.1.9.2

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

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{\frac{1^{2} (- 7 y^{2} + 32)}{2^{2} \cdot 1}}}{4}$

Step 6.2.1.9.3

Rearrange the fraction $\frac{1^{2} (- 7 y^{2} + 32)}{2^{2} \cdot 1}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \sqrt{{(\frac{1}{2})}^{2} (- 7 y^{2} + 32)}}{4}$

Step 6.2.1.10

Pull terms out from under the radical.

$f (\frac{y}{2}) = \frac{\frac{y}{2} - (\frac{1}{2} \cdot \sqrt{- 7 y^{2} + 32})}{4}$

Step 6.2.1.11

Combine $\frac{1}{2}$ and $\sqrt{- 7 y^{2} + 32}$ .

$f (\frac{y}{2}) = \frac{\frac{y}{2} - \frac{\sqrt{- 7 y^{2} + 32}}{2}}{4}$

Step 6.2.1.12

Combine the numerators over the common denominator.

$f (\frac{y}{2}) = \frac{\frac{y - \sqrt{- 7 y^{2} + 32}}{2}}{4}$

Step 6.2.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{y}{2}) = \frac{y - \sqrt{- 7 y^{2} + 32}}{2} \cdot \frac{1}{4}$

Step 6.2.3

Multiply $\frac{y - \sqrt{- 7 y^{2} + 32}}{2} \cdot \frac{1}{4}$ .

Tap for more steps...

Step 6.2.3.1

Multiply $\frac{y - \sqrt{- 7 y^{2} + 32}}{2}$ by $\frac{1}{4}$ .

$f (\frac{y}{2}) = \frac{y - \sqrt{- 7 y^{2} + 32}}{2 \cdot 4}$

Step 6.2.3.2

Multiply $2$ by $4$ .

$f (\frac{y}{2}) = \frac{y - \sqrt{- 7 y^{2} + 32}}{8}$

Step 6.2.4

The final answer is $\frac{y - \sqrt{- 7 y^{2} + 32}}{8}$ .

$\frac{y - \sqrt{- 7 y^{2} + 32}}{8}$

Step 7

The horizontal tangent lines are $y = \frac{y + \sqrt{- 7 y^{2} + 32}}{8}, y = \frac{y - \sqrt{- 7 y^{2} + 32}}{8}$

$y = \frac{y + \sqrt{- 7 y^{2} + 32}}{8}, y = \frac{y - \sqrt{- 7 y^{2} + 32}}{8}$

Step 8