Calculus Examples

$y = \frac{5}{x^{2} - 8 x - 63}$

Step 1

Write $y = \frac{5}{x^{2} - 8 x - 63}$ as a function.

$f (x) = \frac{5}{x^{2} - 8 x - 63}$

Step 2

Find the first derivative of the function.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{2} - 8 x - 63}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{2} - 8 x - 63}]$ .

$5 \frac{d}{d x} [\frac{1}{x^{2} - 8 x - 63}]$

Step 2.1.2

Rewrite $\frac{1}{x^{2} - 8 x - 63}$ as ${(x^{2} - 8 x - 63)}^{- 1}$ .

$5 \frac{d}{d x} [{(x^{2} - 8 x - 63)}^{- 1}]$

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - 8 x - 63$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 8 x - 63$ .

$5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 2.2.2

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

$5 (- u^{- 2} \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} - 8 x - 63$ .

$5 (- {(x^{2} - 8 x - 63)}^{- 2} \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $5$ .

$- 5 ({(x^{2} - 8 x - 63)}^{- 2} \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 2.3.2

By the Sum Rule, the derivative of $x^{2} - 8 x - 63$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 63]$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 63])$

Step 2.3.3

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

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 63])$

Step 2.3.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [- 63])$

Step 2.3.5

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

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 \cdot 1 + \frac{d}{d x} [- 63])$

Step 2.3.6

Multiply $- 8$ by $1$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 + \frac{d}{d x} [- 63])$

Step 2.3.7

Since $- 63$ is constant with respect to $x$ , the derivative of $- 63$ with respect to $x$ is $0$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 + 0)$

Step 2.3.8

Add $2 x - 8$ and $0$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8)$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 \frac{1}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $- 5$ and $\frac{1}{{(x^{2} - 8 x - 63)}^{2}}$ .

$\frac{- 5}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$

Step 2.4.2.2

Move the negative in front of the fraction.

$- \frac{5}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$

Step 2.4.3

Reorder the factors of $- \frac{5}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$ .

$- (2 x - 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.4

Apply the distributive property.

$(- (2 x) - - 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.5

Multiply $2$ by $- 1$ .

$(- 2 x - - 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.6

Multiply $- 1$ by $- 8$ .

$(- 2 x + 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.7

Multiply $- 2 x + 8$ by $\frac{5}{{(x^{2} - 8 x - 63)}^{2}}$ .

$\frac{(- 2 x + 8) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.8

Simplify the numerator.

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Step 2.4.8.1

Factor $2$ out of $- 2 x + 8$ .

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Step 2.4.8.1.1

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

$\frac{(2 (- x) + 8) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.8.1.2

Factor $2$ out of $8$ .

$\frac{(2 (- x) + 2 (4)) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.8.1.3

Factor $2$ out of $2 (- x) + 2 (4)$ .

$\frac{2 (- x + 4) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.8.2

Multiply $5$ by $2$ .

$\frac{10 (- x + 4)}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.9

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

$\frac{10 (- (x) + 4)}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.10

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

$\frac{10 (- (x) - 1 (- 4))}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.11

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

$\frac{10 (- (x - 4))}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.12

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

$\frac{10 (- 1 (x - 4))}{{(x^{2} - 8 x - 63)}^{2}}$

Step 2.4.13

Move the negative in front of the fraction.

$- \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}}$ with respect to $x$ is $- 10 \frac{d}{d x} [\frac{x - 4}{{(x^{2} - 8 x - 63)}^{2}}]$ .

$f'' (x) = - 10 \frac{d}{d x} \frac{x - 4}{{(x^{2} - 8 x - 63)}^{2}}$

Step 3.2

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 $f (x) = x - 4$ and $g (x) = {(x^{2} - 8 x - 63)}^{2}$ .

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} \frac{d}{d x} (x - 4) - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{({(x^{2} - 8 x - 63)}^{2})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(x^{2} - 8 x - 63)}^{2})}^{2}$ .

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Step 3.3.1.1

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

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} \frac{d}{d x} (x - 4) - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{(x^{2} - 8 x - 63)}^{2 \cdot 2}}$

Step 3.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} \frac{d}{d x} (x - 4) - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.3.2

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

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 4)) - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.3.3

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

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} (1 + \frac{d}{d x} (- 4)) - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.3.4

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

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} (1 + 0) - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} \cdot 1 - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.3.5.2

Multiply ${(x^{2} - 8 x - 63)}^{2}$ by $1$ .

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} - (x - 4) \frac{d}{d x} {(x^{2} - 8 x - 63)}^{2}}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.4

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 $g (x) = x^{2} - 8 x - 63$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{2} - 8 x - 63$ .

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} - (x - 4) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} - 8 x - 63))}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.4.2

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

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} - (x - 4) (2 u \frac{d}{d x} (x^{2} - 8 x - 63))}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.4.3

Replace all occurrences of $u$ with $x^{2} - 8 x - 63$ .

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} - (x - 4) (2 (x^{2} - 8 x - 63) \frac{d}{d x} (x^{2} - 8 x - 63))}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.5

Simplify with factoring out.

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Step 3.5.1

Multiply $2$ by $- 1$ .

$f'' (x) = - 10 \frac{{(x^{2} - 8 x - 63)}^{2} - 2 (x - 4) ((x^{2} - 8 x - 63) \frac{d}{d x} (x^{2} - 8 x - 63))}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.5.2

Factor $x^{2} - 8 x - 63$ out of ${(x^{2} - 8 x - 63)}^{2} - 2 (x - 4) ((x^{2} - 8 x - 63) \frac{d}{d x} [x^{2} - 8 x - 63])$ .

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Step 3.5.2.1

Factor $x^{2} - 8 x - 63$ out of ${(x^{2} - 8 x - 63)}^{2}$ .

$f'' (x) = - 10 \frac{(x^{2} - 8 x - 63) (x^{2} - 8 x - 63) - 2 (x - 4) ((x^{2} - 8 x - 63) \frac{d}{d x} (x^{2} - 8 x - 63))}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.5.2.2

Factor $x^{2} - 8 x - 63$ out of $- 2 (x - 4) ((x^{2} - 8 x - 63) \frac{d}{d x} [x^{2} - 8 x - 63])$ .

$f'' (x) = - 10 \frac{(x^{2} - 8 x - 63) (x^{2} - 8 x - 63) + (x^{2} - 8 x - 63) (- 2 (x - 4) (\frac{d}{d x} (x^{2} - 8 x - 63)))}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.5.2.3

Factor $x^{2} - 8 x - 63$ out of $(x^{2} - 8 x - 63) (x^{2} - 8 x - 63) + (x^{2} - 8 x - 63) (- 2 (x - 4) (\frac{d}{d x} [x^{2} - 8 x - 63]))$ .

$f'' (x) = - 10 \frac{(x^{2} - 8 x - 63) (x^{2} - 8 x - 63 - 2 (x - 4) (\frac{d}{d x} (x^{2} - 8 x - 63)))}{{(x^{2} - 8 x - 63)}^{4}}$

Step 3.6

Cancel the common factors.

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Step 3.6.1

Factor $x^{2} - 8 x - 63$ out of ${(x^{2} - 8 x - 63)}^{4}$ .

$f'' (x) = - 10 \frac{(x^{2} - 8 x - 63) (x^{2} - 8 x - 63 - 2 (x - 4) (\frac{d}{d x} (x^{2} - 8 x - 63)))}{(x^{2} - 8 x - 63) {(x^{2} - 8 x - 63)}^{3}}$

Step 3.6.2

Cancel the common factor.

$f'' (x) = - 10 \frac{(x^{2} - 8 x - 63) (x^{2} - 8 x - 63 - 2 (x - 4) (\frac{d}{d x} (x^{2} - 8 x - 63)))}{(x^{2} - 8 x - 63) {(x^{2} - 8 x - 63)}^{3}}$

Step 3.6.3

Rewrite the expression.

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (\frac{d}{d x} (x^{2} - 8 x - 63))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.7

By the Sum Rule, the derivative of $x^{2} - 8 x - 63$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 63]$ .

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 63))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.8

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

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (2 x + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 63))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.9

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8 \frac{d}{d x} x + \frac{d}{d x} (- 63))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.10

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

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8 \cdot 1 + \frac{d}{d x} (- 63))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.11

Multiply $- 8$ by $1$ .

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8 + \frac{d}{d x} (- 63))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.12

Since $- 63$ is constant with respect to $x$ , the derivative of $- 63$ with respect to $x$ is $0$ .

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8 + 0)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.13

Combine fractions.

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Step 3.13.1

Add $2 x - 8$ and $0$ .

$f'' (x) = - 10 \frac{x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.13.2

Combine $- 10$ and $\frac{x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8)}{{(x^{2} - 8 x - 63)}^{3}}$ .

$f'' (x) = \frac{- 10 (x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.13.3

Move the negative in front of the fraction.

$f'' (x) = - \frac{10 (x^{2} - 8 x - 63 - 2 (x - 4) (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14

Simplify.

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Step 3.14.1

Apply the distributive property.

$f'' (x) = - \frac{10 (x^{2} - 8 x - 63 + (- 2 x - 2 \cdot - 4) (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.2

Apply the distributive property.

$f'' (x) = - \frac{10 x^{2} + 10 (- 8 x) + 10 \cdot - 63 + 10 ((- 2 x - 2 \cdot - 4) (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3

Simplify the numerator.

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Step 3.14.3.1

Simplify each term.

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Step 3.14.3.1.1

Multiply $- 8$ by $10$ .

$f'' (x) = - \frac{10 x^{2} - 80 x + 10 \cdot - 63 + 10 ((- 2 x - 2 \cdot - 4) (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.2

Multiply $10$ by $- 63$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 ((- 2 x - 2 \cdot - 4) (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.3

Multiply $- 2$ by $- 4$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 ((- 2 x + 8) (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.4

Expand $(- 2 x + 8) (2 x - 8)$ using the FOIL Method.

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Step 3.14.3.1.4.1

Apply the distributive property.

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 2 x (2 x - 8) + 8 (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.4.2

Apply the distributive property.

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 2 x (2 x) - 2 x \cdot - 8 + 8 (2 x - 8))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.4.3

Apply the distributive property.

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 2 x (2 x) - 2 x \cdot - 8 + 8 (2 x) + 8 \cdot - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5

Simplify and combine like terms.

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Step 3.14.3.1.5.1

Simplify each term.

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Step 3.14.3.1.5.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 2 \cdot (2 x \cdot x) - 2 x \cdot - 8 + 8 (2 x) + 8 \cdot - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5.1.2

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

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Step 3.14.3.1.5.1.2.1

Move $x$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 2 \cdot (2 (x \cdot x)) - 2 x \cdot - 8 + 8 (2 x) + 8 \cdot - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 2 \cdot (2 x^{2}) - 2 x \cdot - 8 + 8 (2 x) + 8 \cdot - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5.1.3

Multiply $- 2$ by $2$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 4 x^{2} - 2 x \cdot - 8 + 8 (2 x) + 8 \cdot - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5.1.4

Multiply $- 8$ by $- 2$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 4 x^{2} + 16 x + 8 (2 x) + 8 \cdot - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5.1.5

Multiply $2$ by $8$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 4 x^{2} + 16 x + 16 x + 8 \cdot - 8)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5.1.6

Multiply $8$ by $- 8$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 4 x^{2} + 16 x + 16 x - 64)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.5.2

Add $16 x$ and $16 x$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 4 x^{2} + 32 x - 64)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.6

Apply the distributive property.

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 + 10 (- 4 x^{2}) + 10 (32 x) + 10 \cdot - 64}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.7

Simplify.

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Step 3.14.3.1.7.1

Multiply $- 4$ by $10$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 - 40 x^{2} + 10 (32 x) + 10 \cdot - 64}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.7.2

Multiply $32$ by $10$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 - 40 x^{2} + 320 x + 10 \cdot - 64}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.1.7.3

Multiply $10$ by $- 64$ .

$f'' (x) = - \frac{10 x^{2} - 80 x - 630 - 40 x^{2} + 320 x - 640}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.2

Subtract $40 x^{2}$ from $10 x^{2}$ .

$f'' (x) = - \frac{- 30 x^{2} - 80 x - 630 + 320 x - 640}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.3

Add $- 80 x$ and $320 x$ .

$f'' (x) = - \frac{- 30 x^{2} + 240 x - 630 - 640}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.3.4

Subtract $640$ from $- 630$ .

$f'' (x) = - \frac{- 30 x^{2} + 240 x - 1270}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.4

Factor $10$ out of $- 30 x^{2} + 240 x - 1270$ .

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Step 3.14.4.1

Factor $10$ out of $- 30 x^{2}$ .

$f'' (x) = - \frac{10 (- 3 x^{2}) + 240 x - 1270}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.4.2

Factor $10$ out of $240 x$ .

$f'' (x) = - \frac{10 (- 3 x^{2}) + 10 (24 x) - 1270}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.4.3

Factor $10$ out of $- 1270$ .

$f'' (x) = - \frac{10 (- 3 x^{2}) + 10 (24 x) + 10 (- 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.4.4

Factor $10$ out of $10 (- 3 x^{2}) + 10 (24 x)$ .

$f'' (x) = - \frac{10 (- 3 x^{2} + 24 x) + 10 (- 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.4.5

Factor $10$ out of $10 (- 3 x^{2} + 24 x) + 10 (- 127)$ .

$f'' (x) = - \frac{10 (- 3 x^{2} + 24 x - 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.5

Factor $- 1$ out of $- 3 x^{2}$ .

$f'' (x) = - \frac{10 (- (3 x^{2}) + 24 x - 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.6

Factor $- 1$ out of $24 x$ .

$f'' (x) = - \frac{10 (- (3 x^{2}) - (- 24 x) - 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.7

Factor $- 1$ out of $- (3 x^{2}) - (- 24 x)$ .

$f'' (x) = - \frac{10 (- (3 x^{2} - 24 x) - 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.8

Rewrite $- 127$ as $- 1 (127)$ .

$f'' (x) = - \frac{10 (- (3 x^{2} - 24 x) - 1 \cdot 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.9

Factor $- 1$ out of $- (3 x^{2} - 24 x) - 1 (127)$ .

$f'' (x) = - \frac{10 (- (3 x^{2} - 24 x + 127))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.10

Rewrite $- (3 x^{2} - 24 x + 127)$ as $- 1 (3 x^{2} - 24 x + 127)$ .

$f'' (x) = - \frac{10 (- 1 (3 x^{2} - 24 x + 127))}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.11

Move the negative in front of the fraction.

$f'' (x) = \frac{10 (3 x^{2} - 24 x + 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 3.14.12

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{10 (3 x^{2} - 24 x + 127)}{{(x^{2} - 8 x - 63)}^{3}})$

Step 3.14.13

Multiply $\frac{10 (3 x^{2} - 24 x + 127)}{{(x^{2} - 8 x - 63)}^{3}}$ by $1$ .

$f'' (x) = \frac{10 (3 x^{2} - 24 x + 127)}{{(x^{2} - 8 x - 63)}^{3}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Differentiate using the Constant Multiple Rule.

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Step 5.1.1.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{2} - 8 x - 63}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{2} - 8 x - 63}]$ .

$5 \frac{d}{d x} [\frac{1}{x^{2} - 8 x - 63}]$

Step 5.1.1.2

Rewrite $\frac{1}{x^{2} - 8 x - 63}$ as ${(x^{2} - 8 x - 63)}^{- 1}$ .

$5 \frac{d}{d x} [{(x^{2} - 8 x - 63)}^{- 1}]$

Step 5.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} - 8 x - 63$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 8 x - 63$ .

$5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 5.1.2.2

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

$5 (- u^{- 2} \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 5.1.2.3

Replace all occurrences of $u$ with $x^{2} - 8 x - 63$ .

$5 (- {(x^{2} - 8 x - 63)}^{- 2} \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 5.1.3

Differentiate.

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Step 5.1.3.1

Multiply $- 1$ by $5$ .

$- 5 ({(x^{2} - 8 x - 63)}^{- 2} \frac{d}{d x} [x^{2} - 8 x - 63])$

Step 5.1.3.2

By the Sum Rule, the derivative of $x^{2} - 8 x - 63$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 63]$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 63])$

Step 5.1.3.3

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

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 63])$

Step 5.1.3.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [- 63])$

Step 5.1.3.5

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

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 \cdot 1 + \frac{d}{d x} [- 63])$

Step 5.1.3.6

Multiply $- 8$ by $1$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 + \frac{d}{d x} [- 63])$

Step 5.1.3.7

Since $- 63$ is constant with respect to $x$ , the derivative of $- 63$ with respect to $x$ is $0$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8 + 0)$

Step 5.1.3.8

Add $2 x - 8$ and $0$ .

$- 5 {(x^{2} - 8 x - 63)}^{- 2} (2 x - 8)$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 \frac{1}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$

Step 5.1.4.2

Combine terms.

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Step 5.1.4.2.1

Combine $- 5$ and $\frac{1}{{(x^{2} - 8 x - 63)}^{2}}$ .

$\frac{- 5}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$

Step 5.1.4.2.2

Move the negative in front of the fraction.

$- \frac{5}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$

Step 5.1.4.3

Reorder the factors of $- \frac{5}{{(x^{2} - 8 x - 63)}^{2}} (2 x - 8)$ .

$- (2 x - 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.4

Apply the distributive property.

$(- (2 x) - - 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.5

Multiply $2$ by $- 1$ .

$(- 2 x - - 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.6

Multiply $- 1$ by $- 8$ .

$(- 2 x + 8) \frac{5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.7

Multiply $- 2 x + 8$ by $\frac{5}{{(x^{2} - 8 x - 63)}^{2}}$ .

$\frac{(- 2 x + 8) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.8

Simplify the numerator.

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Step 5.1.4.8.1

Factor $2$ out of $- 2 x + 8$ .

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Step 5.1.4.8.1.1

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

$\frac{(2 (- x) + 8) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.8.1.2

Factor $2$ out of $8$ .

$\frac{(2 (- x) + 2 (4)) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.8.1.3

Factor $2$ out of $2 (- x) + 2 (4)$ .

$\frac{2 (- x + 4) \cdot 5}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.8.2

Multiply $5$ by $2$ .

$\frac{10 (- x + 4)}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.9

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

$\frac{10 (- (x) + 4)}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.10

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

$\frac{10 (- (x) - 1 (- 4))}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.11

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

$\frac{10 (- (x - 4))}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.12

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

$\frac{10 (- 1 (x - 4))}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.1.4.13

Move the negative in front of the fraction.

$f' (x) = - \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}}$ .

$- \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}} = 0$

Step 6.2

Set the numerator equal to zero.

$10 (x - 4) = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $10 (x - 4) = 0$ by $10$ and simplify.

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Step 6.3.1.1

Divide each term in $10 (x - 4) = 0$ by $10$ .

$\frac{10 (x - 4)}{10} = \frac{0}{10}$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $10$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

$\frac{10 (x - 4)}{10} = \frac{0}{10}$

Step 6.3.1.2.1.2

Divide $x - 4$ by $1$ .

$x - 4 = \frac{0}{10}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $10$ .

$x - 4 = 0$

Step 6.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{10 (x - 4)}{{(x^{2} - 8 x - 63)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 8 x - 63)}^{2} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x^{2} - 8 x - 63 = \pm \sqrt{0}$

Step 7.2.2

Simplify $\pm \sqrt{0}$ .

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Step 7.2.2.1

Rewrite $0$ as $0^{2}$ .

$x^{2} - 8 x - 63 = \pm \sqrt{0^{2}}$

Step 7.2.2.2

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

$x^{2} - 8 x - 63 = \pm 0$

Step 7.2.2.3

Plus or minus $0$ is $0$ .

$x^{2} - 8 x - 63 = 0$

Step 7.2.3

Use the quadratic formula to find the solutions.

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

Step 7.2.4

Substitute the values $a = 1$ , $b = - 8$ , and $c = - 63$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 63)}}{2 \cdot 1}$

Step 7.2.5

Simplify.

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Step 7.2.5.1

Simplify the numerator.

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Step 7.2.5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 63}}{2 \cdot 1}$

Step 7.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 63$ .

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Step 7.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 63}}{2 \cdot 1}$

Step 7.2.5.1.2.2

Multiply $- 4$ by $- 63$ .

$x = \frac{8 \pm \sqrt{64 + 252}}{2 \cdot 1}$

Step 7.2.5.1.3

Add $64$ and $252$ .

$x = \frac{8 \pm \sqrt{316}}{2 \cdot 1}$

Step 7.2.5.1.4

Rewrite $316$ as $2^{2} \cdot 79$ .

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Step 7.2.5.1.4.1

Factor $4$ out of $316$ .

$x = \frac{8 \pm \sqrt{4 (79)}}{2 \cdot 1}$

Step 7.2.5.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 79}}{2 \cdot 1}$

Step 7.2.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{79}}{2 \cdot 1}$

Step 7.2.5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{79}}{2}$

Step 7.2.5.3

Simplify $\frac{8 \pm 2 \sqrt{79}}{2}$ .

$x = 4 \pm \sqrt{79}$

Step 7.2.6

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

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Step 7.2.6.1

Simplify the numerator.

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Step 7.2.6.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 63}}{2 \cdot 1}$

Step 7.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 63$ .

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Step 7.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 63}}{2 \cdot 1}$

Step 7.2.6.1.2.2

Multiply $- 4$ by $- 63$ .

$x = \frac{8 \pm \sqrt{64 + 252}}{2 \cdot 1}$

Step 7.2.6.1.3

Add $64$ and $252$ .

$x = \frac{8 \pm \sqrt{316}}{2 \cdot 1}$

Step 7.2.6.1.4

Rewrite $316$ as $2^{2} \cdot 79$ .

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Step 7.2.6.1.4.1

Factor $4$ out of $316$ .

$x = \frac{8 \pm \sqrt{4 (79)}}{2 \cdot 1}$

Step 7.2.6.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 79}}{2 \cdot 1}$

Step 7.2.6.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{79}}{2 \cdot 1}$

Step 7.2.6.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{79}}{2}$

Step 7.2.6.3

Simplify $\frac{8 \pm 2 \sqrt{79}}{2}$ .

$x = 4 \pm \sqrt{79}$

Step 7.2.6.4

Change the $\pm$ to $+$ .

$x = 4 + \sqrt{79}$

Step 7.2.7

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

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Step 7.2.7.1

Simplify the numerator.

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Step 7.2.7.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 63}}{2 \cdot 1}$

Step 7.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 63$ .

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Step 7.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 63}}{2 \cdot 1}$

Step 7.2.7.1.2.2

Multiply $- 4$ by $- 63$ .

$x = \frac{8 \pm \sqrt{64 + 252}}{2 \cdot 1}$

Step 7.2.7.1.3

Add $64$ and $252$ .

$x = \frac{8 \pm \sqrt{316}}{2 \cdot 1}$

Step 7.2.7.1.4

Rewrite $316$ as $2^{2} \cdot 79$ .

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Step 7.2.7.1.4.1

Factor $4$ out of $316$ .

$x = \frac{8 \pm \sqrt{4 (79)}}{2 \cdot 1}$

Step 7.2.7.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 79}}{2 \cdot 1}$

Step 7.2.7.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{79}}{2 \cdot 1}$

Step 7.2.7.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{79}}{2}$

Step 7.2.7.3

Simplify $\frac{8 \pm 2 \sqrt{79}}{2}$ .

$x = 4 \pm \sqrt{79}$

Step 7.2.7.4

Change the $\pm$ to $-$ .

$x = 4 - \sqrt{79}$

Step 7.2.8

The final answer is the combination of both solutions.

$x = 4 + \sqrt{79}, 4 - \sqrt{79}$

Step 7.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 4 - \sqrt{79}, x = 4 + \sqrt{79}$

Step 8

Critical points to evaluate.

$x = 4$

Step 9

Evaluate the second derivative at $x = 4$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{10 (3 {(4)}^{2} - 24 \cdot 4 + 127)}{{({(4)}^{2} - 8 \cdot 4 - 63)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $4$ to the power of $2$ .

$\frac{10 (3 \cdot 16 - 24 \cdot 4 + 127)}{{(4^{2} - 8 \cdot 4 - 63)}^{3}}$

Step 10.1.2

Multiply $3$ by $16$ .

$\frac{10 (48 - 24 \cdot 4 + 127)}{{(4^{2} - 8 \cdot 4 - 63)}^{3}}$

Step 10.1.3

Multiply $- 24$ by $4$ .

$\frac{10 (48 - 96 + 127)}{{(4^{2} - 8 \cdot 4 - 63)}^{3}}$

Step 10.1.4

Subtract $96$ from $48$ .

$\frac{10 (- 48 + 127)}{{(4^{2} - 8 \cdot 4 - 63)}^{3}}$

Step 10.1.5

Add $- 48$ and $127$ .

$\frac{10 \cdot 79}{{(4^{2} - 8 \cdot 4 - 63)}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Raise $4$ to the power of $2$ .

$\frac{10 \cdot 79}{{(16 - 8 \cdot 4 - 63)}^{3}}$

Step 10.2.2

Multiply $- 8$ by $4$ .

$\frac{10 \cdot 79}{{(16 - 32 - 63)}^{3}}$

Step 10.2.3

Subtract $32$ from $16$ .

$\frac{10 \cdot 79}{{(- 16 - 63)}^{3}}$

Step 10.2.4

Subtract $63$ from $- 16$ .

$\frac{10 \cdot 79}{{(- 79)}^{3}}$

Step 10.2.5

Raise $- 79$ to the power of $3$ .

$\frac{10 \cdot 79}{- 493039}$

Step 10.3

Reduce the expression by cancelling the common factors.

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Step 10.3.1

Multiply $10$ by $79$ .

$\frac{790}{- 493039}$

Step 10.3.2

Cancel the common factor of $790$ and $- 493039$ .

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Step 10.3.2.1

Factor $79$ out of $790$ .

$\frac{79 (10)}{- 493039}$

Step 10.3.2.2

Cancel the common factors.

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Step 10.3.2.2.1

Factor $79$ out of $- 493039$ .

$\frac{79 \cdot 10}{79 \cdot - 6241}$

Step 10.3.2.2.2

Cancel the common factor.

$\frac{79 \cdot 10}{79 \cdot - 6241}$

Step 10.3.2.2.3

Rewrite the expression.

$\frac{10}{- 6241}$

Step 10.3.3

Move the negative in front of the fraction.

$- \frac{10}{6241}$

Step 11

$x = 4$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 4$ is a local maximum

Step 12

Find the y-value when $x = 4$ .

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Step 12.1

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

$f (4) = \frac{5}{{(4)}^{2} - 8 \cdot 4 - 63}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify the denominator.

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Step 12.2.1.1

Raise $4$ to the power of $2$ .

$f (4) = \frac{5}{16 - 8 \cdot 4 - 63}$

Step 12.2.1.2

Multiply $- 8$ by $4$ .

$f (4) = \frac{5}{16 - 32 - 63}$

Step 12.2.1.3

Subtract $32$ from $16$ .

$f (4) = \frac{5}{- 16 - 63}$

Step 12.2.1.4

Subtract $63$ from $- 16$ .

$f (4) = \frac{5}{- 79}$

Step 12.2.2

Move the negative in front of the fraction.

$f (4) = - \frac{5}{79}$

Step 12.2.3

The final answer is $- \frac{5}{79}$ .

$y = - \frac{5}{79}$

Step 13

These are the local extrema for $f (x) = \frac{5}{x^{2} - 8 x - 63}$ .

$(4, - \frac{5}{79})$ is a local maxima

Step 14