Calculus Examples

$f (x) = \frac{x - 1}{x^{2} + 4}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 1$ and $g (x) = x^{2} + 4$ .

$f' (x) = \frac{(x^{2} + 4) \frac{d}{d x} (x - 1) - (x - 1) \frac{d}{d x} (x^{2} + 4)}{{(x^{2} + 4)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$f' (x) = \frac{(x^{2} + 4) (\frac{d}{d x} (x) + \frac{d}{d x} (- 1)) - (x - 1) \frac{d}{d x} (x^{2} + 4)}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.2

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

$f' (x) = \frac{(x^{2} + 4) (1 + \frac{d}{d x} (- 1)) - (x - 1) \frac{d}{d x} (x^{2} + 4)}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.3

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

$f' (x) = \frac{(x^{2} + 4) (1 + 0) - (x - 1) \frac{d}{d x} (x^{2} + 4)}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.4

Simplify the expression.

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Step 1.1.2.4.1

Add $1$ and $0$ .

$f' (x) = \frac{(x^{2} + 4) \cdot 1 - (x - 1) \frac{d}{d x} (x^{2} + 4)}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.4.2

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

$f' (x) = \frac{x^{2} + 4 - (x - 1) \frac{d}{d x} (x^{2} + 4)}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.5

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

$f' (x) = \frac{x^{2} + 4 - (x - 1) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (4))}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.6

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

$f' (x) = \frac{x^{2} + 4 - (x - 1) (2 x + \frac{d}{d x} (4))}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.7

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

$f' (x) = \frac{x^{2} + 4 - (x - 1) (2 x + 0)}{{(x^{2} + 4)}^{2}}$

Step 1.1.2.8

Simplify the expression.

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Step 1.1.2.8.1

Add $2 x$ and $0$ .

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

Step 1.1.2.8.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$f' (x) = \frac{x^{2} + 4 + (- 2 x - 2 \cdot - 1) x}{{(x^{2} + 4)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$f' (x) = \frac{x^{2} + 4 - 2 x \cdot x - 2 \cdot (- 1 x)}{{(x^{2} + 4)}^{2}}$

Step 1.1.3.3

Simplify the numerator.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.3.1.1.1

Move $x$ .

$f' (x) = \frac{x^{2} + 4 - 2 (x \cdot x) - 2 \cdot (- 1 x)}{{(x^{2} + 4)}^{2}}$

Step 1.1.3.3.1.1.2

Multiply $x$ by $x$ .

$f' (x) = \frac{x^{2} + 4 - 2 x^{2} - 2 \cdot (- 1 x)}{{(x^{2} + 4)}^{2}}$

Step 1.1.3.3.1.2

Multiply $- 2$ by $- 1$ .

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

Step 1.1.3.3.2

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

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

Step 1.1.3.4

Reorder terms.

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

Step 1.1.3.5

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

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

Step 1.1.3.6

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

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

Step 1.1.3.7

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

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

Step 1.1.3.8

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

$f' (x) = \frac{- (x^{2} - 2 x) - 1 \cdot - 4}{{(x^{2} + 4)}^{2}}$

Step 1.1.3.9

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

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

Step 1.1.3.10

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

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

Step 1.1.3.11

Move the negative in front of the fraction.

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{x^{2} - 2 x - 4}{{(x^{2} + 4)}^{2}}$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{x^{2} - 2 x - 4}{{(x^{2} + 4)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{x^{2} - 2 x - 4}{{(x^{2} + 4)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

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

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Use the quadratic formula to find the solutions.

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

Step 2.3.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the numerator.

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Step 2.3.3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 2.3.3.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 2.3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 4}}{2 \cdot 1}$

Step 2.3.3.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 2.3.3.1.3

Add $4$ and $16$ .

$x = \frac{2 \pm \sqrt{20}}{2 \cdot 1}$

Step 2.3.3.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.3.3.1.4.1

Factor $4$ out of $20$ .

$x = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 2.3.3.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 2.3.3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 2.3.3.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{5}}{2}$

Step 2.3.3.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{2}$ .

$x = 1 \pm \sqrt{5}$

Step 2.3.4

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

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Step 2.3.4.1

Simplify the numerator.

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Step 2.3.4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 2.3.4.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 2.3.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 4}}{2 \cdot 1}$

Step 2.3.4.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 2.3.4.1.3

Add $4$ and $16$ .

$x = \frac{2 \pm \sqrt{20}}{2 \cdot 1}$

Step 2.3.4.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.3.4.1.4.1

Factor $4$ out of $20$ .

$x = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 2.3.4.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 2.3.4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 2.3.4.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{5}}{2}$

Step 2.3.4.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{2}$ .

$x = 1 \pm \sqrt{5}$

Step 2.3.4.4

Change the $\pm$ to $+$ .

$x = 1 + \sqrt{5}$

Step 2.3.5

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

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Step 2.3.5.1

Simplify the numerator.

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Step 2.3.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 2.3.5.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 2.3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 4}}{2 \cdot 1}$

Step 2.3.5.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 2.3.5.1.3

Add $4$ and $16$ .

$x = \frac{2 \pm \sqrt{20}}{2 \cdot 1}$

Step 2.3.5.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.3.5.1.4.1

Factor $4$ out of $20$ .

$x = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 2.3.5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 2.3.5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 2.3.5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 \sqrt{5}}{2}$

Step 2.3.5.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{2}$ .

$x = 1 \pm \sqrt{5}$

Step 2.3.5.4

Change the $\pm$ to $-$ .

$x = 1 - \sqrt{5}$

Step 2.3.6

The final answer is the combination of both solutions.

$x = 1 + \sqrt{5}, 1 - \sqrt{5}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $\frac{x - 1}{x^{2} + 4}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 1 + \sqrt{5}$ .

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Step 4.1.1

Substitute $1 + \sqrt{5}$ for $x$ .

$\frac{(1 + \sqrt{5}) - 1}{{(1 + \sqrt{5})}^{2} + 4}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Subtract $1$ from $1$ .

$\frac{0 + \sqrt{5}}{{(1 + \sqrt{5})}^{2} + 4}$

Step 4.1.2.1.2

Add $0$ and $\sqrt{5}$ .

$\frac{\sqrt{5}}{{(1 + \sqrt{5})}^{2} + 4}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Rewrite ${(1 + \sqrt{5})}^{2}$ as $(1 + \sqrt{5}) (1 + \sqrt{5})$ .

$\frac{\sqrt{5}}{(1 + \sqrt{5}) (1 + \sqrt{5}) + 4}$

Step 4.1.2.2.2

Expand $(1 + \sqrt{5}) (1 + \sqrt{5})$ using the FOIL Method.

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Step 4.1.2.2.2.1

Apply the distributive property.

$\frac{\sqrt{5}}{1 (1 + \sqrt{5}) + \sqrt{5} (1 + \sqrt{5}) + 4}$

Step 4.1.2.2.2.2

Apply the distributive property.

$\frac{\sqrt{5}}{1 \cdot 1 + 1 \sqrt{5} + \sqrt{5} (1 + \sqrt{5}) + 4}$

Step 4.1.2.2.2.3

Apply the distributive property.

$\frac{\sqrt{5}}{1 \cdot 1 + 1 \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5} + 4}$

Step 4.1.2.2.3

Simplify and combine like terms.

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Step 4.1.2.2.3.1

Simplify each term.

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Step 4.1.2.2.3.1.1

Multiply $1$ by $1$ .

$\frac{\sqrt{5}}{1 + 1 \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5} + 4}$

Step 4.1.2.2.3.1.2

Multiply $\sqrt{5}$ by $1$ .

$\frac{\sqrt{5}}{1 + \sqrt{5} + \sqrt{5} \cdot 1 + \sqrt{5} \sqrt{5} + 4}$

Step 4.1.2.2.3.1.3

Multiply $\sqrt{5}$ by $1$ .

$\frac{\sqrt{5}}{1 + \sqrt{5} + \sqrt{5} + \sqrt{5} \sqrt{5} + 4}$

Step 4.1.2.2.3.1.4

Combine using the product rule for radicals.

$\frac{\sqrt{5}}{1 + \sqrt{5} + \sqrt{5} + \sqrt{5 \cdot 5} + 4}$

Step 4.1.2.2.3.1.5

Multiply $5$ by $5$ .

$\frac{\sqrt{5}}{1 + \sqrt{5} + \sqrt{5} + \sqrt{25} + 4}$

Step 4.1.2.2.3.1.6

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

$\frac{\sqrt{5}}{1 + \sqrt{5} + \sqrt{5} + \sqrt{5^{2}} + 4}$

Step 4.1.2.2.3.1.7

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

$\frac{\sqrt{5}}{1 + \sqrt{5} + \sqrt{5} + 5 + 4}$

Step 4.1.2.2.3.2

Add $1$ and $5$ .

$\frac{\sqrt{5}}{6 + \sqrt{5} + \sqrt{5} + 4}$

Step 4.1.2.2.3.3

Add $\sqrt{5}$ and $\sqrt{5}$ .

$\frac{\sqrt{5}}{6 + 2 \sqrt{5} + 4}$

Step 4.1.2.2.4

Add $6$ and $4$ .

$\frac{\sqrt{5}}{10 + 2 \sqrt{5}}$

Step 4.1.2.3

Multiply $\frac{\sqrt{5}}{10 + 2 \sqrt{5}}$ by $\frac{10 - 2 \sqrt{5}}{10 - 2 \sqrt{5}}$ .

$\frac{\sqrt{5}}{10 + 2 \sqrt{5}} \cdot \frac{10 - 2 \sqrt{5}}{10 - 2 \sqrt{5}}$

Step 4.1.2.4

Multiply $\frac{\sqrt{5}}{10 + 2 \sqrt{5}}$ by $\frac{10 - 2 \sqrt{5}}{10 - 2 \sqrt{5}}$ .

$\frac{\sqrt{5} (10 - 2 \sqrt{5})}{(10 + 2 \sqrt{5}) (10 - 2 \sqrt{5})}$

Step 4.1.2.5

Expand the denominator using the FOIL method.

$\frac{\sqrt{5} (10 - 2 \sqrt{5})}{100 - 20 \sqrt{5} + 20 \sqrt{5} - 4 {\sqrt{5}}^{2}}$

Step 4.1.2.6

Simplify.

$\frac{\sqrt{5} (10 - 2 \sqrt{5})}{80}$

Step 4.1.2.7

Cancel the common factor of $10 - 2 \sqrt{5}$ and $80$ .

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Step 4.1.2.7.1

Factor $2$ out of $\sqrt{5} (10 - 2 \sqrt{5})$ .

$\frac{2 (\sqrt{5} (5 - \sqrt{5}))}{80}$

Step 4.1.2.7.2

Cancel the common factors.

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Step 4.1.2.7.2.1

Factor $2$ out of $80$ .

$\frac{2 (\sqrt{5} (5 - \sqrt{5}))}{2 (40)}$

Step 4.1.2.7.2.2

Cancel the common factor.

$\frac{2 (\sqrt{5} (5 - \sqrt{5}))}{2 \cdot 40}$

Step 4.1.2.7.2.3

Rewrite the expression.

$\frac{\sqrt{5} (5 - \sqrt{5})}{40}$

Step 4.1.2.8

Apply the distributive property.

$\frac{\sqrt{5} \cdot 5 + \sqrt{5} (- \sqrt{5})}{40}$

Step 4.1.2.9

Move $5$ to the left of $\sqrt{5}$ .

$\frac{5 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{5})}{40}$

Step 4.1.2.10

Multiply $\sqrt{5} (- \sqrt{5})$ .

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Step 4.1.2.10.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5 \cdot \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})}{40}$

Step 4.1.2.10.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5 \cdot \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{40}$

Step 4.1.2.10.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 \cdot \sqrt{5} - {\sqrt{5}}^{1 + 1}}{40}$

Step 4.1.2.10.4

Add $1$ and $1$ .

$\frac{5 \cdot \sqrt{5} - {\sqrt{5}}^{2}}{40}$

Step 4.1.2.11

Simplify each term.

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Step 4.1.2.11.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1.2.11.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5 \sqrt{5} - {(5^{\frac{1}{2}})}^{2}}{40}$

Step 4.1.2.11.1.2

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

$\frac{5 \sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{40}$

Step 4.1.2.11.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5 \sqrt{5} - 5^{\frac{2}{2}}}{40}$

Step 4.1.2.11.1.4

Cancel the common factor of $2$ .

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Step 4.1.2.11.1.4.1

Cancel the common factor.

$\frac{5 \sqrt{5} - 5^{\frac{2}{2}}}{40}$

Step 4.1.2.11.1.4.2

Rewrite the expression.

$\frac{5 \sqrt{5} - 5^{1}}{40}$

Step 4.1.2.11.1.5

Evaluate the exponent.

$\frac{5 \sqrt{5} - 1 \cdot 5}{40}$

Step 4.1.2.11.2

Multiply $- 1$ by $5$ .

$\frac{5 \sqrt{5} - 5}{40}$

Step 4.1.2.12

Cancel the common factor of $5 \sqrt{5} - 5$ and $40$ .

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Step 4.1.2.12.1

Factor $5$ out of $5 \sqrt{5}$ .

$\frac{5 (\sqrt{5}) - 5}{40}$

Step 4.1.2.12.2

Factor $5$ out of $- 5$ .

$\frac{5 (\sqrt{5}) + 5 (- 1)}{40}$

Step 4.1.2.12.3

Factor $5$ out of $5 (\sqrt{5}) + 5 (- 1)$ .

$\frac{5 (\sqrt{5} - 1)}{40}$

Step 4.1.2.12.4

Cancel the common factors.

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Step 4.1.2.12.4.1

Factor $5$ out of $40$ .

$\frac{5 (\sqrt{5} - 1)}{5 (8)}$

Step 4.1.2.12.4.2

Cancel the common factor.

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

Step 4.1.2.12.4.3

Rewrite the expression.

$\frac{\sqrt{5} - 1}{8}$

Step 4.2

Evaluate at $x = 1 - \sqrt{5}$ .

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Step 4.2.1

Substitute $1 - \sqrt{5}$ for $x$ .

$\frac{(1 - \sqrt{5}) - 1}{{(1 - \sqrt{5})}^{2} + 4}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Subtract $1$ from $1$ .

$\frac{0 - \sqrt{5}}{{(1 - \sqrt{5})}^{2} + 4}$

Step 4.2.2.1.2

Subtract $\sqrt{5}$ from $0$ .

$\frac{- \sqrt{5}}{{(1 - \sqrt{5})}^{2} + 4}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Rewrite ${(1 - \sqrt{5})}^{2}$ as $(1 - \sqrt{5}) (1 - \sqrt{5})$ .

$\frac{- \sqrt{5}}{(1 - \sqrt{5}) (1 - \sqrt{5}) + 4}$

Step 4.2.2.2.2

Expand $(1 - \sqrt{5}) (1 - \sqrt{5})$ using the FOIL Method.

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Step 4.2.2.2.2.1

Apply the distributive property.

$\frac{- \sqrt{5}}{1 (1 - \sqrt{5}) - \sqrt{5} (1 - \sqrt{5}) + 4}$

Step 4.2.2.2.2.2

Apply the distributive property.

$\frac{- \sqrt{5}}{1 \cdot 1 + 1 (- \sqrt{5}) - \sqrt{5} (1 - \sqrt{5}) + 4}$

Step 4.2.2.2.2.3

Apply the distributive property.

$\frac{- \sqrt{5}}{1 \cdot 1 + 1 (- \sqrt{5}) - \sqrt{5} \cdot 1 - \sqrt{5} (- \sqrt{5}) + 4}$

Step 4.2.2.2.3

Simplify and combine like terms.

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Step 4.2.2.2.3.1

Simplify each term.

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Step 4.2.2.2.3.1.1

Multiply $1$ by $1$ .

$\frac{- \sqrt{5}}{1 + 1 (- \sqrt{5}) - \sqrt{5} \cdot 1 - \sqrt{5} (- \sqrt{5}) + 4}$

Step 4.2.2.2.3.1.2

Multiply $- \sqrt{5}$ by $1$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} \cdot 1 - \sqrt{5} (- \sqrt{5}) + 4}$

Step 4.2.2.2.3.1.3

Multiply $- 1$ by $1$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} - \sqrt{5} (- \sqrt{5}) + 4}$

Step 4.2.2.2.3.1.4

Multiply $- \sqrt{5} (- \sqrt{5})$ .

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Step 4.2.2.2.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + 1 \sqrt{5} \sqrt{5} + 4}$

Step 4.2.2.2.3.1.4.2

Multiply $\sqrt{5}$ by $1$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + \sqrt{5} \sqrt{5} + 4}$

Step 4.2.2.2.3.1.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5} + 4}$

Step 4.2.2.2.3.1.4.4

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1} + 4}$

Step 4.2.2.2.3.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{1 + 1} + 4}$

Step 4.2.2.2.3.1.4.6

Add $1$ and $1$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + {\sqrt{5}}^{2} + 4}$

Step 4.2.2.2.3.1.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.2.2.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + {(5^{\frac{1}{2}})}^{2} + 4}$

Step 4.2.2.2.3.1.5.2

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

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + 5^{\frac{1}{2} \cdot 2} + 4}$

Step 4.2.2.2.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + 5^{\frac{2}{2}} + 4}$

Step 4.2.2.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.3.1.5.4.1

Cancel the common factor.

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + 5^{\frac{2}{2}} + 4}$

Step 4.2.2.2.3.1.5.4.2

Rewrite the expression.

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + 5^{1} + 4}$

Step 4.2.2.2.3.1.5.5

Evaluate the exponent.

$\frac{- \sqrt{5}}{1 - \sqrt{5} - \sqrt{5} + 5 + 4}$

Step 4.2.2.2.3.2

Add $1$ and $5$ .

$\frac{- \sqrt{5}}{6 - \sqrt{5} - \sqrt{5} + 4}$

Step 4.2.2.2.3.3

Subtract $\sqrt{5}$ from $- \sqrt{5}$ .

$\frac{- \sqrt{5}}{6 - 2 \sqrt{5} + 4}$

Step 4.2.2.2.4

Add $6$ and $4$ .

$\frac{- \sqrt{5}}{10 - 2 \sqrt{5}}$

Step 4.2.2.3

Move the negative in front of the fraction.

$- \frac{\sqrt{5}}{10 - 2 \sqrt{5}}$

Step 4.2.2.4

Multiply $\frac{\sqrt{5}}{10 - 2 \sqrt{5}}$ by $\frac{10 + 2 \sqrt{5}}{10 + 2 \sqrt{5}}$ .

$- (\frac{\sqrt{5}}{10 - 2 \sqrt{5}} \cdot \frac{10 + 2 \sqrt{5}}{10 + 2 \sqrt{5}})$

Step 4.2.2.5

Simplify terms.

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Step 4.2.2.5.1

Multiply $\frac{\sqrt{5}}{10 - 2 \sqrt{5}}$ by $\frac{10 + 2 \sqrt{5}}{10 + 2 \sqrt{5}}$ .

$- \frac{\sqrt{5} (10 + 2 \sqrt{5})}{(10 - 2 \sqrt{5}) (10 + 2 \sqrt{5})}$

Step 4.2.2.5.2

Expand the denominator using the FOIL method.

$- \frac{\sqrt{5} (10 + 2 \sqrt{5})}{100 + 20 \sqrt{5} - 20 \sqrt{5} - 4 {\sqrt{5}}^{2}}$

Step 4.2.2.5.3

Simplify.

$- \frac{\sqrt{5} (10 + 2 \sqrt{5})}{80}$

Step 4.2.2.5.4

Cancel the common factor of $10 + 2 \sqrt{5}$ and $80$ .

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Step 4.2.2.5.4.1

Factor $2$ out of $\sqrt{5} (10 + 2 \sqrt{5})$ .

$- \frac{2 (\sqrt{5} (5 + \sqrt{5}))}{80}$

Step 4.2.2.5.4.2

Cancel the common factors.

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Step 4.2.2.5.4.2.1

Factor $2$ out of $80$ .

$- \frac{2 (\sqrt{5} (5 + \sqrt{5}))}{2 (40)}$

Step 4.2.2.5.4.2.2

Cancel the common factor.

$- \frac{2 (\sqrt{5} (5 + \sqrt{5}))}{2 \cdot 40}$

Step 4.2.2.5.4.2.3

Rewrite the expression.

$- \frac{\sqrt{5} (5 + \sqrt{5})}{40}$

Step 4.2.2.5.5

Apply the distributive property.

$- \frac{\sqrt{5} \cdot 5 + \sqrt{5} \sqrt{5}}{40}$

Step 4.2.2.5.6

Move $5$ to the left of $\sqrt{5}$ .

$- \frac{5 \cdot \sqrt{5} + \sqrt{5} \sqrt{5}}{40}$

Step 4.2.2.5.7

Combine using the product rule for radicals.

$- \frac{5 \cdot \sqrt{5} + \sqrt{5 \cdot 5}}{40}$

Step 4.2.2.6

Simplify each term.

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Step 4.2.2.6.1

Multiply $5$ by $5$ .

$- \frac{5 \sqrt{5} + \sqrt{25}}{40}$

Step 4.2.2.6.2

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

$- \frac{5 \sqrt{5} + \sqrt{5^{2}}}{40}$

Step 4.2.2.6.3

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

$- \frac{5 \sqrt{5} + 5}{40}$

Step 4.2.2.7

Cancel the common factor of $5 \sqrt{5} + 5$ and $40$ .

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Step 4.2.2.7.1

Factor $5$ out of $5 \sqrt{5}$ .

$- \frac{5 (\sqrt{5}) + 5}{40}$

Step 4.2.2.7.2

Factor $5$ out of $5$ .

$- \frac{5 (\sqrt{5}) + 5 (1)}{40}$

Step 4.2.2.7.3

Factor $5$ out of $5 (\sqrt{5}) + 5 (1)$ .

$- \frac{5 (\sqrt{5} + 1)}{40}$

Step 4.2.2.7.4

Cancel the common factors.

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Step 4.2.2.7.4.1

Factor $5$ out of $40$ .

$- \frac{5 (\sqrt{5} + 1)}{5 (8)}$

Step 4.2.2.7.4.2

Cancel the common factor.

$- \frac{5 (\sqrt{5} + 1)}{5 \cdot 8}$

Step 4.2.2.7.4.3

Rewrite the expression.

$- \frac{\sqrt{5} + 1}{8}$

Step 4.3

List all of the points.

$(1 + \sqrt{5}, \frac{\sqrt{5} - 1}{8}), (1 - \sqrt{5}, - \frac{\sqrt{5} + 1}{8})$

Step 5