Calculus Examples

$36 x^{2} + 16 y^{2} + 288 x - 192 y + 576 = 0$

Step 1

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

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 16$ , $b = - 192$ , and $c = 36 x^{2} + 288 x + 576$ into the quadratic formula and solve for $y$ .

$\frac{192 \pm \sqrt{{(- 192)}^{2} - 4 \cdot (16 \cdot (36 x^{2} + 288 x + 576))}}{2 \cdot 16}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Rewrite $4 \cdot 16 \cdot (36 x^{2} + 288 x + 576)$ as ${(2 \cdot 4 \cdot (6 x + 24))}^{2}$ .

$y = \frac{192 \pm \sqrt{{(- 192)}^{2} - {(2 \cdot 4 \cdot (6 x + 24))}^{2}}}{2 \cdot 16}$

Step 1.3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 192$ and $b = 2 \cdot 4 \cdot (6 x + 24)$ .

$y = \frac{192 \pm \sqrt{(- 192 + 2 \cdot 4 \cdot (6 x + 24)) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3

Simplify.

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Step 1.3.1.3.1

Multiply $2$ by $4$ .

$y = \frac{192 \pm \sqrt{(- 192 + 8 \cdot (6 x + 24)) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3.2

Apply the distributive property.

$y = \frac{192 \pm \sqrt{(- 192 + 8 (6 x) + 8 \cdot 24) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3.3

Multiply $6$ by $8$ .

$y = \frac{192 \pm \sqrt{(- 192 + 48 x + 8 \cdot 24) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3.4

Multiply $8$ by $24$ .

$y = \frac{192 \pm \sqrt{(- 192 + 48 x + 192) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3.5

Add $- 192$ and $192$ .

$y = \frac{192 \pm \sqrt{(48 x + 0) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3.6

Add $48 x$ and $0$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3.7

Combine exponents.

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Step 1.3.1.3.7.1

Multiply $2$ by $4$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - (8 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.3.1.3.7.2

Multiply $8$ by $- 1$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 8 (6 x + 24))}}{2 \cdot 16}$

Step 1.3.1.4

Simplify each term.

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Step 1.3.1.4.1

Apply the distributive property.

$y = \frac{192 \pm \sqrt{48 x (- 192 - 8 (6 x) - 8 \cdot 24)}}{2 \cdot 16}$

Step 1.3.1.4.2

Multiply $6$ by $- 8$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 48 x - 8 \cdot 24)}}{2 \cdot 16}$

Step 1.3.1.4.3

Multiply $- 8$ by $24$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 48 x - 192)}}{2 \cdot 16}$

Step 1.3.1.5

Subtract $192$ from $- 192$ .

$y = \frac{192 \pm \sqrt{48 x (- 48 x - 384)}}{2 \cdot 16}$

Step 1.3.1.6

Factor $48$ out of $- 48 x - 384$ .

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Step 1.3.1.6.1

Factor $48$ out of $- 48 x$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x) - 384)}}{2 \cdot 16}$

Step 1.3.1.6.2

Factor $48$ out of $- 384$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x) + 48 (- 8))}}{2 \cdot 16}$

Step 1.3.1.6.3

Factor $48$ out of $48 (- x) + 48 (- 8)$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x - 8))}}{2 \cdot 16}$

$y = \frac{192 \pm \sqrt{48 x \cdot (48 (- x - 8))}}{2 \cdot 16}$

Step 1.3.1.7

Multiply $48$ by $48$ .

$y = \frac{192 \pm \sqrt{2304 x (- x - 8)}}{2 \cdot 16}$

Step 1.3.1.8

Rewrite $2304 x (- x - 8)$ as $48^{2} (x (- x - 8))$ .

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Step 1.3.1.8.1

Rewrite $2304$ as $48^{2}$ .

$y = \frac{192 \pm \sqrt{48^{2} x (- x - 8)}}{2 \cdot 16}$

Step 1.3.1.8.2

Add parentheses.

$y = \frac{192 \pm \sqrt{48^{2} (x (- x - 8))}}{2 \cdot 16}$

Step 1.3.1.9

Pull terms out from under the radical.

$y = \frac{192 \pm 48 \sqrt{x (- x - 8)}}{2 \cdot 16}$

Step 1.3.2

Multiply $2$ by $16$ .

$y = \frac{192 \pm 48 \sqrt{x (- x - 8)}}{32}$

Step 1.3.3

Simplify $\frac{192 \pm 48 \sqrt{x (- x - 8)}}{32}$ .

$y = \frac{12 \pm 3 \sqrt{x (- x - 8)}}{2}$

Step 1.4

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

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Rewrite $4 \cdot 16 \cdot (36 x^{2} + 288 x + 576)$ as ${(2 \cdot 4 \cdot (6 x + 24))}^{2}$ .

$y = \frac{192 \pm \sqrt{{(- 192)}^{2} - {(2 \cdot 4 \cdot (6 x + 24))}^{2}}}{2 \cdot 16}$

Step 1.4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 192$ and $b = 2 \cdot 4 \cdot (6 x + 24)$ .

$y = \frac{192 \pm \sqrt{(- 192 + 2 \cdot 4 \cdot (6 x + 24)) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3

Simplify.

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Step 1.4.1.3.1

Multiply $2$ by $4$ .

$y = \frac{192 \pm \sqrt{(- 192 + 8 \cdot (6 x + 24)) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3.2

Apply the distributive property.

$y = \frac{192 \pm \sqrt{(- 192 + 8 (6 x) + 8 \cdot 24) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3.3

Multiply $6$ by $8$ .

$y = \frac{192 \pm \sqrt{(- 192 + 48 x + 8 \cdot 24) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3.4

Multiply $8$ by $24$ .

$y = \frac{192 \pm \sqrt{(- 192 + 48 x + 192) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3.5

Add $- 192$ and $192$ .

$y = \frac{192 \pm \sqrt{(48 x + 0) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3.6

Add $48 x$ and $0$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3.7

Combine exponents.

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Step 1.4.1.3.7.1

Multiply $2$ by $4$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - (8 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.4.1.3.7.2

Multiply $8$ by $- 1$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 8 (6 x + 24))}}{2 \cdot 16}$

Step 1.4.1.4

Simplify each term.

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Step 1.4.1.4.1

Apply the distributive property.

$y = \frac{192 \pm \sqrt{48 x (- 192 - 8 (6 x) - 8 \cdot 24)}}{2 \cdot 16}$

Step 1.4.1.4.2

Multiply $6$ by $- 8$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 48 x - 8 \cdot 24)}}{2 \cdot 16}$

Step 1.4.1.4.3

Multiply $- 8$ by $24$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 48 x - 192)}}{2 \cdot 16}$

Step 1.4.1.5

Subtract $192$ from $- 192$ .

$y = \frac{192 \pm \sqrt{48 x (- 48 x - 384)}}{2 \cdot 16}$

Step 1.4.1.6

Factor $48$ out of $- 48 x - 384$ .

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Step 1.4.1.6.1

Factor $48$ out of $- 48 x$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x) - 384)}}{2 \cdot 16}$

Step 1.4.1.6.2

Factor $48$ out of $- 384$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x) + 48 (- 8))}}{2 \cdot 16}$

Step 1.4.1.6.3

Factor $48$ out of $48 (- x) + 48 (- 8)$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x - 8))}}{2 \cdot 16}$

$y = \frac{192 \pm \sqrt{48 x \cdot (48 (- x - 8))}}{2 \cdot 16}$

Step 1.4.1.7

Multiply $48$ by $48$ .

$y = \frac{192 \pm \sqrt{2304 x (- x - 8)}}{2 \cdot 16}$

Step 1.4.1.8

Rewrite $2304 x (- x - 8)$ as $48^{2} (x (- x - 8))$ .

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Step 1.4.1.8.1

Rewrite $2304$ as $48^{2}$ .

$y = \frac{192 \pm \sqrt{48^{2} x (- x - 8)}}{2 \cdot 16}$

Step 1.4.1.8.2

Add parentheses.

$y = \frac{192 \pm \sqrt{48^{2} (x (- x - 8))}}{2 \cdot 16}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{192 \pm 48 \sqrt{x (- x - 8)}}{2 \cdot 16}$

Step 1.4.2

Multiply $2$ by $16$ .

$y = \frac{192 \pm 48 \sqrt{x (- x - 8)}}{32}$

Step 1.4.3

Simplify $\frac{192 \pm 48 \sqrt{x (- x - 8)}}{32}$ .

$y = \frac{12 \pm 3 \sqrt{x (- x - 8)}}{2}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{12 + 3 \sqrt{x (- x - 8)}}{2}$

Step 1.4.5

Factor $3$ out of $12 + 3 \sqrt{x (- x - 8)}$ .

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Step 1.4.5.1

Factor $3$ out of $12$ .

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

Step 1.4.5.2

Factor $3$ out of $3 \cdot 4 + 3 \sqrt{x (- x - 8)}$ .

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

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Rewrite $4 \cdot 16 \cdot (36 x^{2} + 288 x + 576)$ as ${(2 \cdot 4 \cdot (6 x + 24))}^{2}$ .

$y = \frac{192 \pm \sqrt{{(- 192)}^{2} - {(2 \cdot 4 \cdot (6 x + 24))}^{2}}}{2 \cdot 16}$

Step 1.5.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 192$ and $b = 2 \cdot 4 \cdot (6 x + 24)$ .

$y = \frac{192 \pm \sqrt{(- 192 + 2 \cdot 4 \cdot (6 x + 24)) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3

Simplify.

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Step 1.5.1.3.1

Multiply $2$ by $4$ .

$y = \frac{192 \pm \sqrt{(- 192 + 8 \cdot (6 x + 24)) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3.2

Apply the distributive property.

$y = \frac{192 \pm \sqrt{(- 192 + 8 (6 x) + 8 \cdot 24) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3.3

Multiply $6$ by $8$ .

$y = \frac{192 \pm \sqrt{(- 192 + 48 x + 8 \cdot 24) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3.4

Multiply $8$ by $24$ .

$y = \frac{192 \pm \sqrt{(- 192 + 48 x + 192) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3.5

Add $- 192$ and $192$ .

$y = \frac{192 \pm \sqrt{(48 x + 0) (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3.6

Add $48 x$ and $0$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - (2 \cdot 4 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3.7

Combine exponents.

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Step 1.5.1.3.7.1

Multiply $2$ by $4$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - (8 \cdot (6 x + 24)))}}{2 \cdot 16}$

Step 1.5.1.3.7.2

Multiply $8$ by $- 1$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 8 (6 x + 24))}}{2 \cdot 16}$

Step 1.5.1.4

Simplify each term.

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Step 1.5.1.4.1

Apply the distributive property.

$y = \frac{192 \pm \sqrt{48 x (- 192 - 8 (6 x) - 8 \cdot 24)}}{2 \cdot 16}$

Step 1.5.1.4.2

Multiply $6$ by $- 8$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 48 x - 8 \cdot 24)}}{2 \cdot 16}$

Step 1.5.1.4.3

Multiply $- 8$ by $24$ .

$y = \frac{192 \pm \sqrt{48 x (- 192 - 48 x - 192)}}{2 \cdot 16}$

Step 1.5.1.5

Subtract $192$ from $- 192$ .

$y = \frac{192 \pm \sqrt{48 x (- 48 x - 384)}}{2 \cdot 16}$

Step 1.5.1.6

Factor $48$ out of $- 48 x - 384$ .

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Step 1.5.1.6.1

Factor $48$ out of $- 48 x$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x) - 384)}}{2 \cdot 16}$

Step 1.5.1.6.2

Factor $48$ out of $- 384$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x) + 48 (- 8))}}{2 \cdot 16}$

Step 1.5.1.6.3

Factor $48$ out of $48 (- x) + 48 (- 8)$ .

$y = \frac{192 \pm \sqrt{48 x (48 (- x - 8))}}{2 \cdot 16}$

$y = \frac{192 \pm \sqrt{48 x \cdot (48 (- x - 8))}}{2 \cdot 16}$

Step 1.5.1.7

Multiply $48$ by $48$ .

$y = \frac{192 \pm \sqrt{2304 x (- x - 8)}}{2 \cdot 16}$

Step 1.5.1.8

Rewrite $2304 x (- x - 8)$ as $48^{2} (x (- x - 8))$ .

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Step 1.5.1.8.1

Rewrite $2304$ as $48^{2}$ .

$y = \frac{192 \pm \sqrt{48^{2} x (- x - 8)}}{2 \cdot 16}$

Step 1.5.1.8.2

Add parentheses.

$y = \frac{192 \pm \sqrt{48^{2} (x (- x - 8))}}{2 \cdot 16}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{192 \pm 48 \sqrt{x (- x - 8)}}{2 \cdot 16}$

Step 1.5.2

Multiply $2$ by $16$ .

$y = \frac{192 \pm 48 \sqrt{x (- x - 8)}}{32}$

Step 1.5.3

Simplify $\frac{192 \pm 48 \sqrt{x (- x - 8)}}{32}$ .

$y = \frac{12 \pm 3 \sqrt{x (- x - 8)}}{2}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{12 - 3 \sqrt{x (- x - 8)}}{2}$

Step 1.5.5

Factor $3$ out of $12 - 3 \sqrt{x (- x - 8)}$ .

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Step 1.5.5.1

Factor $3$ out of $12$ .

$y = \frac{3 (4) - 3 \sqrt{x (- x - 8)}}{2}$

Step 1.5.5.2

Factor $3$ out of $- 3 \sqrt{x (- x - 8)}$ .

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

Step 1.5.5.3

Factor $3$ out of $3 (4) + 3 (- \sqrt{x (- x - 8)})$ .

$y = \frac{3 (4 - \sqrt{x (- x - 8)})}{2}$

Step 1.6

The final answer is the combination of both solutions.

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

$y = \frac{3 (4 - \sqrt{x (- x - 8)})}{2}$

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

$y = \frac{3 (4 - \sqrt{x (- x - 8)})}{2}$

Step 2

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

$y = \frac{3 (4 + \sqrt{x (- x - 8)})}{2} \to f (x) = \frac{3 (4 + \sqrt{x (- x - 8)})}{2}$

$y = \frac{3 (4 - \sqrt{x (- x - 8)})}{2} \to f (x) = \frac{3 (4 - \sqrt{x (- x - 8)})}{2}$

Step 3

Because the $y$ variable in the equation $36 x^{2} + 16 y^{2} + 288 x - 192 y + 576 = 0$ 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.

$\frac{d}{d x} (36 x^{2} + 16 y^{2} + 288 x - 192 y + 576) = \frac{d}{d x} (0)$

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 $36 x^{2} + 16 y^{2} + 288 x - 192 y + 576$ with respect to $x$ is $\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [16 y^{2}] + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$ .

$\frac{d}{d x} [36 x^{2}] + \frac{d}{d x} [16 y^{2}] + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.2

Evaluate $\frac{d}{d x} [36 x^{2}]$ .

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Step 3.2.2.1

Since $36$ is constant with respect to $x$ , the derivative of $36 x^{2}$ with respect to $x$ is $36 \frac{d}{d x} [x^{2}]$ .

$36 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [16 y^{2}] + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.2.2

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

$36 (2 x) + \frac{d}{d x} [16 y^{2}] + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.2.3

Multiply $2$ by $36$ .

$72 x + \frac{d}{d x} [16 y^{2}] + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.3

Evaluate $\frac{d}{d x} [16 y^{2}]$ .

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Step 3.2.3.1

Since $16$ is constant with respect to $x$ , the derivative of $16 y^{2}$ with respect to $x$ is $16 \frac{d}{d x} [y^{2}]$ .

$72 x + 16 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

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 $g (x) = y$ .

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Step 3.2.3.2.1

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

$72 x + 16 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

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 $n = 2$ .

$72 x + 16 (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.3.2.3

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

$72 x + 16 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.3.3

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

$72 x + 16 (2 y y') + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.3.4

Multiply $2$ by $16$ .

$72 x + 32 y y' + \frac{d}{d x} [288 x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.4

Evaluate $\frac{d}{d x} [288 x]$ .

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Step 3.2.4.1

Since $288$ is constant with respect to $x$ , the derivative of $288 x$ with respect to $x$ is $288 \frac{d}{d x} [x]$ .

$72 x + 32 y y' + 288 \frac{d}{d x} [x] + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.4.2

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

$72 x + 32 y y' + 288 \cdot 1 + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.4.3

Multiply $288$ by $1$ .

$72 x + 32 y y' + 288 + \frac{d}{d x} [- 192 y] + \frac{d}{d x} [576]$

Step 3.2.5

Evaluate $\frac{d}{d x} [- 192 y]$ .

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Step 3.2.5.1

Since $- 192$ is constant with respect to $x$ , the derivative of $- 192 y$ with respect to $x$ is $- 192 \frac{d}{d x} [y]$ .

$72 x + 32 y y' + 288 - 192 \frac{d}{d x} [y] + \frac{d}{d x} [576]$

Step 3.2.5.2

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

$72 x + 32 y y' + 288 - 192 y' + \frac{d}{d x} [576]$

Step 3.2.6

Since $576$ is constant with respect to $x$ , the derivative of $576$ with respect to $x$ is $0$ .

$72 x + 32 y y' + 288 - 192 y' + 0$

Step 3.2.7

Simplify.

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Step 3.2.7.1

Add $72 x + 32 y y' + 288 - 192 y'$ and $0$ .

$72 x + 32 y y' + 288 - 192 y'$

Step 3.2.7.2

Reorder terms.

$32 y y' + 72 x + 288 - 192 y'$

Step 3.3

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

$0$

Step 3.4

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

$32 y y' + 72 x + 288 - 192 y' = 0$

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 $72 x$ from both sides of the equation.

$32 y y' + 288 - 192 y' = - 72 x$

Step 3.5.1.2

Subtract $288$ from both sides of the equation.

$32 y y' - 192 y' = - 72 x - 288$

Step 3.5.2

Factor $32 y'$ out of $32 y y' - 192 y'$ .

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Step 3.5.2.1

Factor $32 y'$ out of $32 y y'$ .

$32 y' y - 192 y' = - 72 x - 288$

Step 3.5.2.2

Factor $32 y'$ out of $- 192 y'$ .

$32 y' y + 32 y' \cdot - 6 = - 72 x - 288$

Step 3.5.2.3

Factor $32 y'$ out of $32 y' y + 32 y' \cdot - 6$ .

$32 y' (y - 6) = - 72 x - 288$

Step 3.5.3

Divide each term in $32 y' (y - 6) = - 72 x - 288$ by $32 (y - 6)$ and simplify.

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Step 3.5.3.1

Divide each term in $32 y' (y - 6) = - 72 x - 288$ by $32 (y - 6)$ .

$\frac{32 y' (y - 6)}{32 (y - 6)} = \frac{- 72 x}{32 (y - 6)} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $32$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{32 y' (y - 6)}{32 (y - 6)} = \frac{- 72 x}{32 (y - 6)} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.2.1.2

Rewrite the expression.

$\frac{y' (y - 6)}{y - 6} = \frac{- 72 x}{32 (y - 6)} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.2.2

Cancel the common factor of $y - 6$ .

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Step 3.5.3.2.2.1

Cancel the common factor.

$\frac{y' (y - 6)}{y - 6} = \frac{- 72 x}{32 (y - 6)} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 72 x}{32 (y - 6)} + \frac{- 288}{32 (y - 6)}$

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

Cancel the common factor of $- 72$ and $32$ .

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Step 3.5.3.3.1.1.1

Factor $8$ out of $- 72 x$ .

$y' = \frac{8 (- 9 x)}{32 (y - 6)} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.3.1.1.2

Cancel the common factors.

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Step 3.5.3.3.1.1.2.1

Factor $8$ out of $32 (y - 6)$ .

$y' = \frac{8 (- 9 x)}{8 (4 (y - 6))} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{8 (- 9 x)}{8 (4 (y - 6))} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- 9 x}{4 (y - 6)} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{9 x}{4 (y - 6)} + \frac{- 288}{32 (y - 6)}$

Step 3.5.3.3.1.3

Cancel the common factor of $- 288$ and $32$ .

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Step 3.5.3.3.1.3.1

Factor $32$ out of $- 288$ .

$y' = - \frac{9 x}{4 (y - 6)} + \frac{32 \cdot - 9}{32 (y - 6)}$

Step 3.5.3.3.1.3.2

Cancel the common factors.

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Step 3.5.3.3.1.3.2.1

Cancel the common factor.

$y' = - \frac{9 x}{4 (y - 6)} + \frac{32 \cdot - 9}{32 (y - 6)}$

Step 3.5.3.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{9 x}{4 (y - 6)} + \frac{- 9}{y - 6}$

Step 3.5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{9 x}{4 (y - 6)} - \frac{9}{y - 6}$

Step 3.5.3.3.2

To write $- \frac{9}{y - 6}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y' = - \frac{9 x}{4 (y - 6)} - \frac{9}{y - 6} \cdot \frac{4}{4}$

Step 3.5.3.3.3

Write each expression with a common denominator of $4 (y - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.3.3.3.1

Multiply $\frac{9}{y - 6}$ by $\frac{4}{4}$ .

$y' = - \frac{9 x}{4 (y - 6)} - \frac{9 \cdot 4}{(y - 6) \cdot 4}$

Step 3.5.3.3.3.2

Reorder the factors of $(y - 6) \cdot 4$ .

$y' = - \frac{9 x}{4 (y - 6)} - \frac{9 \cdot 4}{4 (y - 6)}$

Step 3.5.3.3.4

Combine the numerators over the common denominator.

$y' = \frac{- 9 x - 9 \cdot 4}{4 (y - 6)}$

Step 3.5.3.3.5

Simplify the numerator.

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Step 3.5.3.3.5.1

Factor $9$ out of $- 9 x - 9 \cdot 4$ .

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Step 3.5.3.3.5.1.1

Factor $9$ out of $- 9 x$ .

$y' = \frac{9 (- x) - 9 \cdot 4}{4 (y - 6)}$

Step 3.5.3.3.5.1.2

Factor $9$ out of $- 9 \cdot 4$ .

$y' = \frac{9 (- x) + 9 (- 1 \cdot 4)}{4 (y - 6)}$

Step 3.5.3.3.5.1.3

Factor $9$ out of $9 (- x) + 9 (- 1 \cdot 4)$ .

$y' = \frac{9 (- x - 1 \cdot 4)}{4 (y - 6)}$

Step 3.5.3.3.5.2

Multiply $- 1$ by $4$ .

$y' = \frac{9 (- x - 4)}{4 (y - 6)}$

Step 3.5.3.3.6

Simplify with factoring out.

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Step 3.5.3.3.6.1

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

$y' = \frac{9 (- (x) - 4)}{4 (y - 6)}$

Step 3.5.3.3.6.2

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

$y' = \frac{9 (- (x) - 1 (4))}{4 (y - 6)}$

Step 3.5.3.3.6.3

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

$y' = \frac{9 (- (x + 4))}{4 (y - 6)}$

Step 3.5.3.3.6.4

Simplify the expression.

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Step 3.5.3.3.6.4.1

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

$y' = \frac{9 (- 1 (x + 4))}{4 (y - 6)}$

Step 3.5.3.3.6.4.2

Move the negative in front of the fraction.

$y' = - \frac{9 (x + 4)}{4 (y - 6)}$

Step 3.6

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

$\frac{d y}{d x} = - \frac{9 (x + 4)}{4 (y - 6)}$

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{9 (x + 4)}{4 (y - 6)} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$9 (x + 4) = 0$

Step 4.2

Solve the equation for $x$ .

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Step 4.2.1

Divide each term in $9 (x + 4) = 0$ by $9$ and simplify.

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Step 4.2.1.1

Divide each term in $9 (x + 4) = 0$ by $9$ .

$\frac{9 (x + 4)}{9} = \frac{0}{9}$

Step 4.2.1.2

Simplify the left side.

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Step 4.2.1.2.1

Cancel the common factor of $9$ .

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Step 4.2.1.2.1.1

Cancel the common factor.

$\frac{9 (x + 4)}{9} = \frac{0}{9}$

Step 4.2.1.2.1.2

Divide $x + 4$ by $1$ .

$x + 4 = \frac{0}{9}$

Step 4.2.1.3

Simplify the right side.

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Step 4.2.1.3.1

Divide $0$ by $9$ .

$x + 4 = 0$

Step 4.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

Solve the function $f (x) = \frac{3 (4 + \sqrt{x (- x - 8)})}{2}$ at $x = - 4$ .

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

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

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

Multiply $- 1$ by $- 4$ .

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

Step 5.2.1.2

Subtract $8$ from $4$ .

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

Step 5.2.1.3

Multiply $- 4$ by $- 4$ .

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

Step 5.2.1.4

Rewrite $16$ as $4^{2}$ .

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

Step 5.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 4) = \frac{3 (4 + 4)}{2}$

Step 5.2.1.6

Add $4$ and $4$ .

$f (- 4) = \frac{3 \cdot 8}{2}$

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Multiply $3$ by $8$ .

$f (- 4) = \frac{24}{2}$

Step 5.2.2.2

Divide $24$ by $2$ .

$f (- 4) = 12$

Step 5.2.3

The final answer is $12$ .

$12$

Step 6

Solve the function $f (x) = \frac{3 (4 - \sqrt{x (- x - 8)})}{2}$ at $x = - 4$ .

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Step 6.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \frac{3 (4 - \sqrt{(- 4) (- (- 4) - 8)})}{2}$

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

Multiply $- 1$ by $- 4$ .

$f (- 4) = \frac{3 (4 - \sqrt{- 4 (4 - 8)})}{2}$

Step 6.2.1.2

Subtract $8$ from $4$ .

$f (- 4) = \frac{3 (4 - \sqrt{- 4 \cdot - 4})}{2}$

Step 6.2.1.3

Multiply $- 4$ by $- 4$ .

$f (- 4) = \frac{3 (4 - \sqrt{16})}{2}$

Step 6.2.1.4

Rewrite $16$ as $4^{2}$ .

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

Step 6.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 4) = \frac{3 (4 - 1 \cdot 4)}{2}$

Step 6.2.1.6

Multiply $- 1$ by $4$ .

$f (- 4) = \frac{3 (4 - 4)}{2}$

Step 6.2.1.7

Subtract $4$ from $4$ .

$f (- 4) = \frac{3 \cdot 0}{2}$

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Multiply $3$ by $0$ .

$f (- 4) = \frac{0}{2}$

Step 6.2.2.2

Divide $0$ by $2$ .

$f (- 4) = 0$

Step 6.2.3

The final answer is $0$ .

$0$

Step 7

The horizontal tangent lines are $y = 12, y = 0$

$y = 12, y = 0$

Step 8