Calculus Examples

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

Step 1

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

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Step 1.1

Subtract $4$ from both sides of the equation.

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

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 $a = 1$ , $b = - 4$ , and $c = x^{2} - 2 x - 4$ into the quadratic formula and solve for $y$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

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

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Step 1.4.1.1.1

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

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

Step 1.4.1.1.2

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

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

Step 1.4.1.1.3

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

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

Step 1.4.1.2

Multiply $x^{2} - 2 x - 4$ by $1$ .

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

Step 1.4.1.3

Subtract $4$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (x^{2} - 2 x - 8)}}{2 \cdot 1}$

Step 1.4.1.4

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 1.4.1.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 1.4.1.4.2

Write the factored form using these integers.

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

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

Step 1.4.1.5

Rewrite $- 4 (x - 4) (x + 2)$ as $2^{2} \cdot (- 1 (x - 4) (x + 2))$ .

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Step 1.4.1.5.1

Factor $4$ out of $- 4$ .

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

Step 1.4.1.5.2

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

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

Step 1.4.1.5.3

Add parentheses.

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

Step 1.4.1.5.4

Add parentheses.

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

Step 1.4.1.6

Pull terms out from under the radical.

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

Step 1.4.2

Multiply $2$ by $1$ .

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

Step 1.4.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (x - 4) (x + 2)}}{2}$ .

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

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

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

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Step 1.5.1.1.1

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

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

Step 1.5.1.1.2

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

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

Step 1.5.1.1.3

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

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

Step 1.5.1.2

Multiply $x^{2} - 2 x - 4$ by $1$ .

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

Step 1.5.1.3

Subtract $4$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (x^{2} - 2 x - 8)}}{2 \cdot 1}$

Step 1.5.1.4

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 1.5.1.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 1.5.1.4.2

Write the factored form using these integers.

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

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

Step 1.5.1.5

Rewrite $- 4 (x - 4) (x + 2)$ as $2^{2} \cdot (- 1 (x - 4) (x + 2))$ .

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Step 1.5.1.5.1

Factor $4$ out of $- 4$ .

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

Step 1.5.1.5.2

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

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

Step 1.5.1.5.3

Add parentheses.

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

Step 1.5.1.5.4

Add parentheses.

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

Step 1.5.1.6

Pull terms out from under the radical.

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

Step 1.5.2

Multiply $2$ by $1$ .

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

Step 1.5.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (x - 4) (x + 2)}}{2}$ .

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

Step 1.5.4

Change the $\pm$ to $+$ .

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

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

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

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Step 1.6.1.1.1

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

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

Step 1.6.1.1.2

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

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

Step 1.6.1.1.3

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

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

Step 1.6.1.2

Multiply $x^{2} - 2 x - 4$ by $1$ .

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

Step 1.6.1.3

Subtract $4$ from $- 4$ .

$y = \frac{4 \pm \sqrt{- 4 (x^{2} - 2 x - 8)}}{2 \cdot 1}$

Step 1.6.1.4

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 1.6.1.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 1.6.1.4.2

Write the factored form using these integers.

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

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

Step 1.6.1.5

Rewrite $- 4 (x - 4) (x + 2)$ as $2^{2} \cdot (- 1 (x - 4) (x + 2))$ .

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Step 1.6.1.5.1

Factor $4$ out of $- 4$ .

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

Step 1.6.1.5.2

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

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

Step 1.6.1.5.3

Add parentheses.

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

Step 1.6.1.5.4

Add parentheses.

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

Step 1.6.1.6

Pull terms out from under the radical.

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

Step 1.6.2

Multiply $2$ by $1$ .

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

Step 1.6.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (x - 4) (x + 2)}}{2}$ .

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

Step 1.6.4

Change the $\pm$ to $-$ .

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

Step 1.7

The final answer is the combination of both solutions.

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

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

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

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

Step 2

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

$y = 2 + \sqrt{- 1 (x - 4) (x + 2)} \to f (x) = 2 + \sqrt{- 1 (x - 4) (x + 2)}$

$y = 2 - \sqrt{- 1 (x - 4) (x + 2)} \to f (x) = 2 - \sqrt{- 1 (x - 4) (x + 2)}$

Step 3

Because the $y$ variable in the equation $x^{2} + y^{2} - 2 x - 4 y = 4$ 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.

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

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

Differentiate.

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Step 3.2.1.1

By the Sum Rule, the derivative of $x^{2} + y^{2} - 2 x - 4 y$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4 y]$ .

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

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 $n = 2$ .

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

Step 3.2.2

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

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Step 3.2.2.1

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 3.2.2.1.1

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

$2 x + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4 y]$

Step 3.2.2.1.2

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

$2 x + 2 u \frac{d}{d x} [y] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4 y]$

Step 3.2.2.1.3

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

$2 x + 2 y \frac{d}{d x} [y] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4 y]$

Step 3.2.2.2

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

$2 x + 2 y y' + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 4 y]$

Step 3.2.3

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

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Step 3.2.3.1

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

$2 x + 2 y y' - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 4 y]$

Step 3.2.3.2

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

$2 x + 2 y y' - 2 \cdot 1 + \frac{d}{d x} [- 4 y]$

Step 3.2.3.3

Multiply $- 2$ by $1$ .

$2 x + 2 y y' - 2 + \frac{d}{d x} [- 4 y]$

Step 3.2.4

Evaluate $\frac{d}{d x} [- 4 y]$ .

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Step 3.2.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 y$ with respect to $x$ is $- 4 \frac{d}{d x} [y]$ .

$2 x + 2 y y' - 2 - 4 \frac{d}{d x} [y]$

Step 3.2.4.2

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

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

Step 3.2.5

Reorder terms.

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

Step 3.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0$

Step 3.4

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

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

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 $2 x$ from both sides of the equation.

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

Step 3.5.1.2

Add $2$ to both sides of the equation.

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

Step 3.5.2

Factor $2 y'$ out of $2 y y' - 4 y'$ .

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Step 3.5.2.1

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

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

Step 3.5.2.2

Factor $2 y'$ out of $- 4 y'$ .

$2 y' y + 2 y' \cdot - 2 = - 2 x + 2$

Step 3.5.2.3

Factor $2 y'$ out of $2 y' y + 2 y' \cdot - 2$ .

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

Step 3.5.3

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

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Step 3.5.3.1

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

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

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $2$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

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

Step 3.5.3.2.1.2

Rewrite the expression.

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

Step 3.5.3.2.2

Cancel the common factor of $y - 2$ .

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Step 3.5.3.2.2.1

Cancel the common factor.

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

Step 3.5.3.2.2.2

Divide $y'$ by $1$ .

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

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 $- 2$ and $2$ .

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Step 3.5.3.3.1.1.1

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

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

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

Cancel the common factor.

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

Step 3.5.3.3.1.1.2.2

Rewrite the expression.

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

Step 3.5.3.3.1.2

Move the negative in front of the fraction.

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

Step 3.5.3.3.1.3

Cancel the common factor of $2$ .

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Step 3.5.3.3.1.3.1

Cancel the common factor.

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

Step 3.5.3.3.1.3.2

Rewrite the expression.

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

Step 3.5.3.3.2

Simplify terms.

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Step 3.5.3.3.2.1

Combine the numerators over the common denominator.

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

Step 3.5.3.3.2.2

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

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

Step 3.5.3.3.2.3

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

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

Step 3.5.3.3.2.4

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

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

Step 3.5.3.3.2.5

Simplify the expression.

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Step 3.5.3.3.2.5.1

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

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

Step 3.5.3.3.2.5.2

Move the negative in front of the fraction.

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

Step 3.6

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

$\frac{d y}{d x} = - \frac{x - 1}{y - 2}$

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{x - 1}{y - 2} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$x - 1 = 0$

Step 4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5

Solve the function $f (x) = 2 + \sqrt{- 1 (x - 4) (x + 2)}$ at $x = 1$ .

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Step 5.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 + \sqrt{- 1 ((1) - 4) ((1) + 2)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Subtract $4$ from $1$ .

$f (1) = 2 + \sqrt{- 1 \cdot (- 3 (1 + 2))}$

Step 5.2.1.2

Multiply $- 1$ by $- 3$ .

$f (1) = 2 + \sqrt{3 (1 + 2)}$

Step 5.2.1.3

Add $1$ and $2$ .

$f (1) = 2 + \sqrt{3 \cdot 3}$

Step 5.2.1.4

Multiply $3$ by $3$ .

$f (1) = 2 + \sqrt{9}$

Step 5.2.1.5

Rewrite $9$ as $3^{2}$ .

$f (1) = 2 + \sqrt{3^{2}}$

Step 5.2.1.6

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

$f (1) = 2 + 3$

Step 5.2.2

Add $2$ and $3$ .

$f (1) = 5$

Step 5.2.3

The final answer is $5$ .

$5$

Step 6

Solve the function $f (x) = 2 - \sqrt{- 1 (x - 4) (x + 2)}$ at $x = 1$ .

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 - \sqrt{- 1 ((1) - 4) ((1) + 2)}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Subtract $4$ from $1$ .

$f (1) = 2 - \sqrt{- 1 \cdot (- 3 (1 + 2))}$

Step 6.2.1.2

Multiply $- 1$ by $- 3$ .

$f (1) = 2 - \sqrt{3 (1 + 2)}$

Step 6.2.1.3

Add $1$ and $2$ .

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

Step 6.2.1.4

Multiply $3$ by $3$ .

$f (1) = 2 - \sqrt{9}$

Step 6.2.1.5

Rewrite $9$ as $3^{2}$ .

$f (1) = 2 - \sqrt{3^{2}}$

Step 6.2.1.6

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

$f (1) = 2 - 1 \cdot 3$

Step 6.2.1.7

Multiply $- 1$ by $3$ .

$f (1) = 2 - 3$

Step 6.2.2

Subtract $3$ from $2$ .

$f (1) = - 1$

Step 6.2.3

The final answer is $- 1$ .

$- 1$

Step 7

The horizontal tangent lines are $y = 5, y = - 1$

$y = 5, y = - 1$

Step 8