Calculus Examples

$64 x^{2} + 25 y^{2} + 640 x - 400 y + 1600 = 0$

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 $a = 25$ , $b = - 400$ , and $c = 64 x^{2} + 640 x + 1600$ into the quadratic formula and solve for $y$ .

$\frac{400 \pm \sqrt{{(- 400)}^{2} - 4 \cdot (25 \cdot (64 x^{2} + 640 x + 1600))}}{2 \cdot 25}$

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 $4 \cdot 25 \cdot (64 x^{2} + 640 x + 1600)$ as ${(2 \cdot 5 \cdot (8 x + 40))}^{2}$ .

$y = \frac{400 \pm \sqrt{{(- 400)}^{2} - {(2 \cdot 5 \cdot (8 x + 40))}^{2}}}{2 \cdot 25}$

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 $a = - 400$ and $b = 2 \cdot 5 \cdot (8 x + 40)$ .

$y = \frac{400 \pm \sqrt{(- 400 + 2 \cdot 5 \cdot (8 x + 40)) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3

Simplify.

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Step 1.3.1.3.1

Multiply $2$ by $5$ .

$y = \frac{400 \pm \sqrt{(- 400 + 10 \cdot (8 x + 40)) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3.2

Apply the distributive property.

$y = \frac{400 \pm \sqrt{(- 400 + 10 (8 x) + 10 \cdot 40) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3.3

Multiply $8$ by $10$ .

$y = \frac{400 \pm \sqrt{(- 400 + 80 x + 10 \cdot 40) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3.4

Multiply $10$ by $40$ .

$y = \frac{400 \pm \sqrt{(- 400 + 80 x + 400) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3.5

Add $- 400$ and $400$ .

$y = \frac{400 \pm \sqrt{(80 x + 0) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3.6

Add $80 x$ and $0$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3.7

Combine exponents.

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Step 1.3.1.3.7.1

Multiply $2$ by $5$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - (10 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.3.1.3.7.2

Multiply $10$ by $- 1$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 10 (8 x + 40))}}{2 \cdot 25}$

Step 1.3.1.4

Simplify each term.

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Step 1.3.1.4.1

Apply the distributive property.

$y = \frac{400 \pm \sqrt{80 x (- 400 - 10 (8 x) - 10 \cdot 40)}}{2 \cdot 25}$

Step 1.3.1.4.2

Multiply $8$ by $- 10$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 80 x - 10 \cdot 40)}}{2 \cdot 25}$

Step 1.3.1.4.3

Multiply $- 10$ by $40$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 80 x - 400)}}{2 \cdot 25}$

Step 1.3.1.5

Subtract $400$ from $- 400$ .

$y = \frac{400 \pm \sqrt{80 x (- 80 x - 800)}}{2 \cdot 25}$

Step 1.3.1.6

Factor $80$ out of $- 80 x - 800$ .

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Step 1.3.1.6.1

Factor $80$ out of $- 80 x$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x) - 800)}}{2 \cdot 25}$

Step 1.3.1.6.2

Factor $80$ out of $- 800$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x) + 80 (- 10))}}{2 \cdot 25}$

Step 1.3.1.6.3

Factor $80$ out of $80 (- x) + 80 (- 10)$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x - 10))}}{2 \cdot 25}$

$y = \frac{400 \pm \sqrt{80 x \cdot (80 (- x - 10))}}{2 \cdot 25}$

Step 1.3.1.7

Multiply $80$ by $80$ .

$y = \frac{400 \pm \sqrt{6400 x (- x - 10)}}{2 \cdot 25}$

Step 1.3.1.8

Rewrite $6400 x (- x - 10)$ as $80^{2} (x (- x - 10))$ .

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Step 1.3.1.8.1

Rewrite $6400$ as $80^{2}$ .

$y = \frac{400 \pm \sqrt{80^{2} x (- x - 10)}}{2 \cdot 25}$

Step 1.3.1.8.2

Add parentheses.

$y = \frac{400 \pm \sqrt{80^{2} (x (- x - 10))}}{2 \cdot 25}$

Step 1.3.1.9

Pull terms out from under the radical.

$y = \frac{400 \pm 80 \sqrt{x (- x - 10)}}{2 \cdot 25}$

Step 1.3.2

Multiply $2$ by $25$ .

$y = \frac{400 \pm 80 \sqrt{x (- x - 10)}}{50}$

Step 1.3.3

Simplify $\frac{400 \pm 80 \sqrt{x (- x - 10)}}{50}$ .

$y = \frac{40 \pm 8 \sqrt{x (- x - 10)}}{5}$

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 $4 \cdot 25 \cdot (64 x^{2} + 640 x + 1600)$ as ${(2 \cdot 5 \cdot (8 x + 40))}^{2}$ .

$y = \frac{400 \pm \sqrt{{(- 400)}^{2} - {(2 \cdot 5 \cdot (8 x + 40))}^{2}}}{2 \cdot 25}$

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 $a = - 400$ and $b = 2 \cdot 5 \cdot (8 x + 40)$ .

$y = \frac{400 \pm \sqrt{(- 400 + 2 \cdot 5 \cdot (8 x + 40)) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3

Simplify.

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Step 1.4.1.3.1

Multiply $2$ by $5$ .

$y = \frac{400 \pm \sqrt{(- 400 + 10 \cdot (8 x + 40)) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3.2

Apply the distributive property.

$y = \frac{400 \pm \sqrt{(- 400 + 10 (8 x) + 10 \cdot 40) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3.3

Multiply $8$ by $10$ .

$y = \frac{400 \pm \sqrt{(- 400 + 80 x + 10 \cdot 40) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3.4

Multiply $10$ by $40$ .

$y = \frac{400 \pm \sqrt{(- 400 + 80 x + 400) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3.5

Add $- 400$ and $400$ .

$y = \frac{400 \pm \sqrt{(80 x + 0) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3.6

Add $80 x$ and $0$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3.7

Combine exponents.

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Step 1.4.1.3.7.1

Multiply $2$ by $5$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - (10 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.4.1.3.7.2

Multiply $10$ by $- 1$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 10 (8 x + 40))}}{2 \cdot 25}$

Step 1.4.1.4

Simplify each term.

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Step 1.4.1.4.1

Apply the distributive property.

$y = \frac{400 \pm \sqrt{80 x (- 400 - 10 (8 x) - 10 \cdot 40)}}{2 \cdot 25}$

Step 1.4.1.4.2

Multiply $8$ by $- 10$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 80 x - 10 \cdot 40)}}{2 \cdot 25}$

Step 1.4.1.4.3

Multiply $- 10$ by $40$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 80 x - 400)}}{2 \cdot 25}$

Step 1.4.1.5

Subtract $400$ from $- 400$ .

$y = \frac{400 \pm \sqrt{80 x (- 80 x - 800)}}{2 \cdot 25}$

Step 1.4.1.6

Factor $80$ out of $- 80 x - 800$ .

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Step 1.4.1.6.1

Factor $80$ out of $- 80 x$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x) - 800)}}{2 \cdot 25}$

Step 1.4.1.6.2

Factor $80$ out of $- 800$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x) + 80 (- 10))}}{2 \cdot 25}$

Step 1.4.1.6.3

Factor $80$ out of $80 (- x) + 80 (- 10)$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x - 10))}}{2 \cdot 25}$

$y = \frac{400 \pm \sqrt{80 x \cdot (80 (- x - 10))}}{2 \cdot 25}$

Step 1.4.1.7

Multiply $80$ by $80$ .

$y = \frac{400 \pm \sqrt{6400 x (- x - 10)}}{2 \cdot 25}$

Step 1.4.1.8

Rewrite $6400 x (- x - 10)$ as $80^{2} (x (- x - 10))$ .

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Step 1.4.1.8.1

Rewrite $6400$ as $80^{2}$ .

$y = \frac{400 \pm \sqrt{80^{2} x (- x - 10)}}{2 \cdot 25}$

Step 1.4.1.8.2

Add parentheses.

$y = \frac{400 \pm \sqrt{80^{2} (x (- x - 10))}}{2 \cdot 25}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{400 \pm 80 \sqrt{x (- x - 10)}}{2 \cdot 25}$

Step 1.4.2

Multiply $2$ by $25$ .

$y = \frac{400 \pm 80 \sqrt{x (- x - 10)}}{50}$

Step 1.4.3

Simplify $\frac{400 \pm 80 \sqrt{x (- x - 10)}}{50}$ .

$y = \frac{40 \pm 8 \sqrt{x (- x - 10)}}{5}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{40 + 8 \sqrt{x (- x - 10)}}{5}$

Step 1.4.5

Factor $8$ out of $40 + 8 \sqrt{x (- x - 10)}$ .

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Step 1.4.5.1

Factor $8$ out of $40$ .

$y = \frac{8 \cdot 5 + 8 \sqrt{x (- x - 10)}}{5}$

Step 1.4.5.2

Factor $8$ out of $8 \cdot 5 + 8 \sqrt{x (- x - 10)}$ .

$y = \frac{8 (5 + \sqrt{x (- x - 10)})}{5}$

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 $4 \cdot 25 \cdot (64 x^{2} + 640 x + 1600)$ as ${(2 \cdot 5 \cdot (8 x + 40))}^{2}$ .

$y = \frac{400 \pm \sqrt{{(- 400)}^{2} - {(2 \cdot 5 \cdot (8 x + 40))}^{2}}}{2 \cdot 25}$

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 $a = - 400$ and $b = 2 \cdot 5 \cdot (8 x + 40)$ .

$y = \frac{400 \pm \sqrt{(- 400 + 2 \cdot 5 \cdot (8 x + 40)) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3

Simplify.

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Step 1.5.1.3.1

Multiply $2$ by $5$ .

$y = \frac{400 \pm \sqrt{(- 400 + 10 \cdot (8 x + 40)) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3.2

Apply the distributive property.

$y = \frac{400 \pm \sqrt{(- 400 + 10 (8 x) + 10 \cdot 40) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3.3

Multiply $8$ by $10$ .

$y = \frac{400 \pm \sqrt{(- 400 + 80 x + 10 \cdot 40) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3.4

Multiply $10$ by $40$ .

$y = \frac{400 \pm \sqrt{(- 400 + 80 x + 400) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3.5

Add $- 400$ and $400$ .

$y = \frac{400 \pm \sqrt{(80 x + 0) (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3.6

Add $80 x$ and $0$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - (2 \cdot 5 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3.7

Combine exponents.

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Step 1.5.1.3.7.1

Multiply $2$ by $5$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - (10 \cdot (8 x + 40)))}}{2 \cdot 25}$

Step 1.5.1.3.7.2

Multiply $10$ by $- 1$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 10 (8 x + 40))}}{2 \cdot 25}$

Step 1.5.1.4

Simplify each term.

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Step 1.5.1.4.1

Apply the distributive property.

$y = \frac{400 \pm \sqrt{80 x (- 400 - 10 (8 x) - 10 \cdot 40)}}{2 \cdot 25}$

Step 1.5.1.4.2

Multiply $8$ by $- 10$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 80 x - 10 \cdot 40)}}{2 \cdot 25}$

Step 1.5.1.4.3

Multiply $- 10$ by $40$ .

$y = \frac{400 \pm \sqrt{80 x (- 400 - 80 x - 400)}}{2 \cdot 25}$

Step 1.5.1.5

Subtract $400$ from $- 400$ .

$y = \frac{400 \pm \sqrt{80 x (- 80 x - 800)}}{2 \cdot 25}$

Step 1.5.1.6

Factor $80$ out of $- 80 x - 800$ .

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Step 1.5.1.6.1

Factor $80$ out of $- 80 x$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x) - 800)}}{2 \cdot 25}$

Step 1.5.1.6.2

Factor $80$ out of $- 800$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x) + 80 (- 10))}}{2 \cdot 25}$

Step 1.5.1.6.3

Factor $80$ out of $80 (- x) + 80 (- 10)$ .

$y = \frac{400 \pm \sqrt{80 x (80 (- x - 10))}}{2 \cdot 25}$

$y = \frac{400 \pm \sqrt{80 x \cdot (80 (- x - 10))}}{2 \cdot 25}$

Step 1.5.1.7

Multiply $80$ by $80$ .

$y = \frac{400 \pm \sqrt{6400 x (- x - 10)}}{2 \cdot 25}$

Step 1.5.1.8

Rewrite $6400 x (- x - 10)$ as $80^{2} (x (- x - 10))$ .

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Step 1.5.1.8.1

Rewrite $6400$ as $80^{2}$ .

$y = \frac{400 \pm \sqrt{80^{2} x (- x - 10)}}{2 \cdot 25}$

Step 1.5.1.8.2

Add parentheses.

$y = \frac{400 \pm \sqrt{80^{2} (x (- x - 10))}}{2 \cdot 25}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{400 \pm 80 \sqrt{x (- x - 10)}}{2 \cdot 25}$

Step 1.5.2

Multiply $2$ by $25$ .

$y = \frac{400 \pm 80 \sqrt{x (- x - 10)}}{50}$

Step 1.5.3

Simplify $\frac{400 \pm 80 \sqrt{x (- x - 10)}}{50}$ .

$y = \frac{40 \pm 8 \sqrt{x (- x - 10)}}{5}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{40 - 8 \sqrt{x (- x - 10)}}{5}$

Step 1.5.5

Factor $8$ out of $40 - 8 \sqrt{x (- x - 10)}$ .

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Step 1.5.5.1

Factor $8$ out of $40$ .

$y = \frac{8 (5) - 8 \sqrt{x (- x - 10)}}{5}$

Step 1.5.5.2

Factor $8$ out of $- 8 \sqrt{x (- x - 10)}$ .

$y = \frac{8 (5) + 8 (- \sqrt{x (- x - 10)})}{5}$

Step 1.5.5.3

Factor $8$ out of $8 (5) + 8 (- \sqrt{x (- x - 10)})$ .

$y = \frac{8 (5 - \sqrt{x (- x - 10)})}{5}$

Step 1.6

The final answer is the combination of both solutions.

$y = \frac{8 (5 + \sqrt{x (- x - 10)})}{5}$

$y = \frac{8 (5 - \sqrt{x (- x - 10)})}{5}$

$y = \frac{8 (5 + \sqrt{x (- x - 10)})}{5}$

$y = \frac{8 (5 - \sqrt{x (- x - 10)})}{5}$

Step 2

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

$y = \frac{8 (5 + \sqrt{x (- x - 10)})}{5} \to f (x) = \frac{8 (5 + \sqrt{x (- x - 10)})}{5}$

$y = \frac{8 (5 - \sqrt{x (- x - 10)})}{5} \to f (x) = \frac{8 (5 - \sqrt{x (- x - 10)})}{5}$

Step 3

Because the $y$ variable in the equation $64 x^{2} + 25 y^{2} + 640 x - 400 y + 1600 = 0$ 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.

$\frac{d}{d x} (64 x^{2} + 25 y^{2} + 640 x - 400 y + 1600) = \frac{d}{d x} (0)$

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 $64 x^{2} + 25 y^{2} + 640 x - 400 y + 1600$ with respect to $x$ is $\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [25 y^{2}] + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$ .

$\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [25 y^{2}] + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.2

Evaluate $\frac{d}{d x} [64 x^{2}]$ .

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Step 3.2.2.1

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{2}$ with respect to $x$ is $64 \frac{d}{d x} [x^{2}]$ .

$64 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [25 y^{2}] + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

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 $n = 2$ .

$64 (2 x) + \frac{d}{d x} [25 y^{2}] + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.2.3

Multiply $2$ by $64$ .

$128 x + \frac{d}{d x} [25 y^{2}] + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.3

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

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Step 3.2.3.1

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

$128 x + 25 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

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 $y$ .

$128 x + 25 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

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 $n = 2$ .

$128 x + 25 (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.3.2.3

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

$128 x + 25 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.3.3

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

$128 x + 25 (2 y y') + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.3.4

Multiply $2$ by $25$ .

$128 x + 50 y y' + \frac{d}{d x} [640 x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.4

Evaluate $\frac{d}{d x} [640 x]$ .

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Step 3.2.4.1

Since $640$ is constant with respect to $x$ , the derivative of $640 x$ with respect to $x$ is $640 \frac{d}{d x} [x]$ .

$128 x + 50 y y' + 640 \frac{d}{d x} [x] + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

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 $n = 1$ .

$128 x + 50 y y' + 640 \cdot 1 + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.4.3

Multiply $640$ by $1$ .

$128 x + 50 y y' + 640 + \frac{d}{d x} [- 400 y] + \frac{d}{d x} [1600]$

Step 3.2.5

Evaluate $\frac{d}{d x} [- 400 y]$ .

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Step 3.2.5.1

Since $- 400$ is constant with respect to $x$ , the derivative of $- 400 y$ with respect to $x$ is $- 400 \frac{d}{d x} [y]$ .

$128 x + 50 y y' + 640 - 400 \frac{d}{d x} [y] + \frac{d}{d x} [1600]$

Step 3.2.5.2

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

$128 x + 50 y y' + 640 - 400 y' + \frac{d}{d x} [1600]$

Step 3.2.6

Since $1600$ is constant with respect to $x$ , the derivative of $1600$ with respect to $x$ is $0$ .

$128 x + 50 y y' + 640 - 400 y' + 0$

Step 3.2.7

Simplify.

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Step 3.2.7.1

Add $128 x + 50 y y' + 640 - 400 y'$ and $0$ .

$128 x + 50 y y' + 640 - 400 y'$

Step 3.2.7.2

Reorder terms.

$50 y y' + 128 x + 640 - 400 y'$

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.

$50 y y' + 128 x + 640 - 400 y' = 0$

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 $128 x$ from both sides of the equation.

$50 y y' + 640 - 400 y' = - 128 x$

Step 3.5.1.2

Subtract $640$ from both sides of the equation.

$50 y y' - 400 y' = - 128 x - 640$

Step 3.5.2

Factor $50 y'$ out of $50 y y' - 400 y'$ .

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Step 3.5.2.1

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

$50 y' y - 400 y' = - 128 x - 640$

Step 3.5.2.2

Factor $50 y'$ out of $- 400 y'$ .

$50 y' y + 50 y' \cdot - 8 = - 128 x - 640$

Step 3.5.2.3

Factor $50 y'$ out of $50 y' y + 50 y' \cdot - 8$ .

$50 y' (y - 8) = - 128 x - 640$

Step 3.5.3

Divide each term in $50 y' (y - 8) = - 128 x - 640$ by $50 (y - 8)$ and simplify.

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Step 3.5.3.1

Divide each term in $50 y' (y - 8) = - 128 x - 640$ by $50 (y - 8)$ .

$\frac{50 y' (y - 8)}{50 (y - 8)} = \frac{- 128 x}{50 (y - 8)} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $50$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{50 y' (y - 8)}{50 (y - 8)} = \frac{- 128 x}{50 (y - 8)} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.2.1.2

Rewrite the expression.

$\frac{y' (y - 8)}{y - 8} = \frac{- 128 x}{50 (y - 8)} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.2.2

Cancel the common factor of $y - 8$ .

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Step 3.5.3.2.2.1

Cancel the common factor.

$\frac{y' (y - 8)}{y - 8} = \frac{- 128 x}{50 (y - 8)} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 128 x}{50 (y - 8)} + \frac{- 640}{50 (y - 8)}$

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 $- 128$ and $50$ .

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Step 3.5.3.3.1.1.1

Factor $2$ out of $- 128 x$ .

$y' = \frac{2 (- 64 x)}{50 (y - 8)} + \frac{- 640}{50 (y - 8)}$

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 $2$ out of $50 (y - 8)$ .

$y' = \frac{2 (- 64 x)}{2 (25 (y - 8))} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (- 64 x)}{2 (25 (y - 8))} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- 64 x}{25 (y - 8)} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{64 x}{25 (y - 8)} + \frac{- 640}{50 (y - 8)}$

Step 3.5.3.3.1.3

Cancel the common factor of $- 640$ and $50$ .

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Step 3.5.3.3.1.3.1

Factor $10$ out of $- 640$ .

$y' = - \frac{64 x}{25 (y - 8)} + \frac{10 (- 64)}{50 (y - 8)}$

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

Factor $10$ out of $50 (y - 8)$ .

$y' = - \frac{64 x}{25 (y - 8)} + \frac{10 (- 64)}{10 (5 (y - 8))}$

Step 3.5.3.3.1.3.2.2

Cancel the common factor.

$y' = - \frac{64 x}{25 (y - 8)} + \frac{10 \cdot - 64}{10 (5 (y - 8))}$

Step 3.5.3.3.1.3.2.3

Rewrite the expression.

$y' = - \frac{64 x}{25 (y - 8)} + \frac{- 64}{5 (y - 8)}$

Step 3.5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{64 x}{25 (y - 8)} - \frac{64}{5 (y - 8)}$

Step 3.6

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

$\frac{d y}{d x} = - \frac{64 x}{25 (y - 8)} - \frac{64}{5 (y - 8)}$

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{64 x}{25 (y - 8)} - \frac{64}{5 (y - 8)} = 0$ .

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Step 4.1

Add $\frac{64}{5 (y - 8)}$ to both sides of the equation.

$- \frac{64 x}{25 (y - 8)} = \frac{64}{5 (y - 8)}$

Step 4.2

Divide each term in $- \frac{64 x}{25 (y - 8)} = \frac{64}{5 (y - 8)}$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- \frac{64 x}{25 (y - 8)} = \frac{64}{5 (y - 8)}$ by $- 1$ .

$\frac{- \frac{64 x}{25 (y - 8)}}{- 1} = \frac{\frac{64}{5 (y - 8)}}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{64 x}{25 (y - 8)}}{1} = \frac{\frac{64}{5 (y - 8)}}{- 1}$

Step 4.2.2.2

Divide $\frac{64 x}{25 (y - 8)}$ by $1$ .

$\frac{64 x}{25 (y - 8)} = \frac{\frac{64}{5 (y - 8)}}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative one from the denominator of $\frac{\frac{64}{5 (y - 8)}}{- 1}$ .

$\frac{64 x}{25 (y - 8)} = - 1 \cdot \frac{64}{5 (y - 8)}$

Step 4.2.3.2

Rewrite $- 1 \cdot \frac{64}{5 (y - 8)}$ as $- \frac{64}{5 (y - 8)}$ .

$\frac{64 x}{25 (y - 8)} = - \frac{64}{5 (y - 8)}$

Step 4.3

Multiply both sides by $25 (y - 8)$ .

$\frac{64 x}{25 (y - 8)} (25 (y - 8)) = - \frac{64}{5 (y - 8)} (25 (y - 8))$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Simplify $\frac{64 x}{25 (y - 8)} (25 (y - 8))$ .

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Step 4.4.1.1.1

Rewrite using the commutative property of multiplication.

$25 \frac{64 x}{25 (y - 8)} (y - 8) = - \frac{64}{5 (y - 8)} (25 (y - 8))$

Step 4.4.1.1.2

Cancel the common factor of $25$ .

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Step 4.4.1.1.2.1

Cancel the common factor.

$25 \frac{64 x}{25 (y - 8)} (y - 8) = - \frac{64}{5 (y - 8)} (25 (y - 8))$

Step 4.4.1.1.2.2

Rewrite the expression.

$\frac{64 x}{y - 8} (y - 8) = - \frac{64}{5 (y - 8)} (25 (y - 8))$

Step 4.4.1.1.3

Cancel the common factor of $y - 8$ .

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Step 4.4.1.1.3.1

Cancel the common factor.

$\frac{64 x}{y - 8} (y - 8) = - \frac{64}{5 (y - 8)} (25 (y - 8))$

Step 4.4.1.1.3.2

Rewrite the expression.

$64 x = - \frac{64}{5 (y - 8)} (25 (y - 8))$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $- \frac{64}{5 (y - 8)} (25 (y - 8))$ .

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Step 4.4.2.1.1

Cancel the common factor of $5 (y - 8)$ .

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Step 4.4.2.1.1.1

Move the leading negative in $- \frac{64}{5 (y - 8)}$ into the numerator.

$64 x = \frac{- 64}{5 (y - 8)} (25 (y - 8))$

Step 4.4.2.1.1.2

Factor $5 (y - 8)$ out of $25 (y - 8)$ .

$64 x = \frac{- 64}{5 (y - 8)} (5 (y - 8) (5))$

Step 4.4.2.1.1.3

Cancel the common factor.

$64 x = \frac{- 64}{5 (y - 8)} (5 (y - 8) \cdot 5)$

Step 4.4.2.1.1.4

Rewrite the expression.

$64 x = - 64 \cdot 5$

Step 4.4.2.1.2

Multiply $- 64$ by $5$ .

$64 x = - 320$

Step 4.5

Divide each term in $64 x = - 320$ by $64$ and simplify.

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Step 4.5.1

Divide each term in $64 x = - 320$ by $64$ .

$\frac{64 x}{64} = \frac{- 320}{64}$

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $64$ .

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Step 4.5.2.1.1

Cancel the common factor.

$\frac{64 x}{64} = \frac{- 320}{64}$

Step 4.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 320}{64}$

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Divide $- 320$ by $64$ .

$x = - 5$

Step 5

Solve the function $f (x) = \frac{8 (5 + \sqrt{x (- x - 10)})}{5}$ at $x = - 5$ .

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Step 5.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \frac{8 (5 + \sqrt{(- 5) (- (- 5) - 10)})}{5}$

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 $- 5$ .

$f (- 5) = \frac{8 (5 + \sqrt{- 5 (5 - 10)})}{5}$

Step 5.2.1.2

Subtract $10$ from $5$ .

$f (- 5) = \frac{8 (5 + \sqrt{- 5 \cdot - 5})}{5}$

Step 5.2.1.3

Multiply $- 5$ by $- 5$ .

$f (- 5) = \frac{8 (5 + \sqrt{25})}{5}$

Step 5.2.1.4

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

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

Step 5.2.1.5

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

$f (- 5) = \frac{8 (5 + 5)}{5}$

Step 5.2.1.6

Add $5$ and $5$ .

$f (- 5) = \frac{8 \cdot 10}{5}$

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Multiply $8$ by $10$ .

$f (- 5) = \frac{80}{5}$

Step 5.2.2.2

Divide $80$ by $5$ .

$f (- 5) = 16$

Step 5.2.3

The final answer is $16$ .

$16$

Step 6

Solve the function $f (x) = \frac{8 (5 - \sqrt{x (- x - 10)})}{5}$ at $x = - 5$ .

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Step 6.1

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \frac{8 (5 - \sqrt{(- 5) (- (- 5) - 10)})}{5}$

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 $- 5$ .

$f (- 5) = \frac{8 (5 - \sqrt{- 5 (5 - 10)})}{5}$

Step 6.2.1.2

Subtract $10$ from $5$ .

$f (- 5) = \frac{8 (5 - \sqrt{- 5 \cdot - 5})}{5}$

Step 6.2.1.3

Multiply $- 5$ by $- 5$ .

$f (- 5) = \frac{8 (5 - \sqrt{25})}{5}$

Step 6.2.1.4

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

$f (- 5) = \frac{8 (5 - \sqrt{5^{2}})}{5}$

Step 6.2.1.5

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

$f (- 5) = \frac{8 (5 - 1 \cdot 5)}{5}$

Step 6.2.1.6

Multiply $- 1$ by $5$ .

$f (- 5) = \frac{8 (5 - 5)}{5}$

Step 6.2.1.7

Subtract $5$ from $5$ .

$f (- 5) = \frac{8 \cdot 0}{5}$

Step 6.2.2

Simplify the expression.

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Step 6.2.2.1

Multiply $8$ by $0$ .

$f (- 5) = \frac{0}{5}$

Step 6.2.2.2

Divide $0$ by $5$ .

$f (- 5) = 0$

Step 6.2.3

The final answer is $0$ .

$0$

Step 7

The horizontal tangent lines are $y = 16, y = 0$

$y = 16, y = 0$

Step 8