Calculus Examples

$\frac{12}{x^{2} - 2 x - 3}$

Step 1

Write $\frac{12}{x^{2} - 2 x - 3}$ as a function.

$f (x) = \frac{12}{x^{2} - 2 x - 3}$

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 $12$ is constant with respect to $x$ , the derivative of $\frac{12}{x^{2} - 2 x - 3}$ with respect to $x$ is $12 \frac{d}{d x} [\frac{1}{x^{2} - 2 x - 3}]$ .

$12 \frac{d}{d x} [\frac{1}{x^{2} - 2 x - 3}]$

Step 2.1.2

Rewrite $\frac{1}{x^{2} - 2 x - 3}$ as ${(x^{2} - 2 x - 3)}^{- 1}$ .

$12 \frac{d}{d x} [{(x^{2} - 2 x - 3)}^{- 1}]$

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 $g (x) = x^{2} - 2 x - 3$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 2 x - 3$ .

$12 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 2.2.2

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

$12 (- u^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} - 2 x - 3$ .

$12 (- {(x^{2} - 2 x - 3)}^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $12$ .

$- 12 ({(x^{2} - 2 x - 3)}^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 2.3.2

By the Sum Rule, the derivative of $x^{2} - 2 x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3]$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3])$

Step 2.3.3

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

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3])$

Step 2.3.4

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

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 2.3.5

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

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 \cdot 1 + \frac{d}{d x} [- 3])$

Step 2.3.6

Multiply $- 2$ by $1$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 + \frac{d}{d x} [- 3])$

Step 2.3.7

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 + 0)$

Step 2.3.8

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

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2)$

Step 2.4

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

$- 12 \frac{1}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$

Step 2.5

Simplify.

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Step 2.5.1

Combine terms.

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Step 2.5.1.1

Combine $- 12$ and $\frac{1}{{(x^{2} - 2 x - 3)}^{2}}$ .

$\frac{- 12}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$

Step 2.5.1.2

Move the negative in front of the fraction.

$- \frac{12}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$

Step 2.5.2

Reorder the factors of $- \frac{12}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$ .

$- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ with respect to $x$ is $- \frac{d}{d x} [(2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}}]$ .

$f'' (x) = - \frac{d}{d x} ((2 x - 2) (\frac{12}{{(x^{2} - 2 x - 3)}^{2}}))$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 2 x - 2$ and $g (x) = \frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ .

$f'' (x) = - ((2 x - 2) \frac{d}{d x} (\frac{12}{{(x^{2} - 2 x - 3)}^{2}}) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.3

Differentiate using the Constant Multiple Rule.

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Step 3.3.1

Since $12$ is constant with respect to $x$ , the derivative of $\frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ with respect to $x$ is $12 \frac{d}{d x} [\frac{1}{{(x^{2} - 2 x - 3)}^{2}}]$ .

$f'' (x) = - ((2 x - 2) (12 \frac{d}{d x} (\frac{1}{{(x^{2} - 2 x - 3)}^{2}})) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Move $12$ to the left of $2 x - 2$ .

$f'' (x) = - (12 \cdot (2 x - 2) \frac{d}{d x} (\frac{1}{{(x^{2} - 2 x - 3)}^{2}}) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.3.2.2

Rewrite $\frac{1}{{(x^{2} - 2 x - 3)}^{2}}$ as ${({(x^{2} - 2 x - 3)}^{2})}^{- 1}$ .

$f'' (x) = - (12 (2 x - 2) \frac{d}{d x} ({({(x^{2} - 2 x - 3)}^{2})}^{- 1}) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.3.2.3

Multiply the exponents in ${({(x^{2} - 2 x - 3)}^{2})}^{- 1}$ .

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Step 3.3.2.3.1

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

$f'' (x) = - (12 (2 x - 2) \frac{d}{d x} ({(x^{2} - 2 x - 3)}^{2 \cdot - 1}) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.3.2.3.2

Multiply $2$ by $- 1$ .

$f'' (x) = - (12 (2 x - 2) \frac{d}{d x} ({(x^{2} - 2 x - 3)}^{- 2}) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

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 $g (x) = x^{2} - 2 x - 3$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{2} - 2 x - 3$ .

$f'' (x) = - (12 (2 x - 2) (\frac{d}{d u} (u^{- 2}) \frac{d}{d x} (x^{2} - 2 x - 3)) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.4.2

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

$f'' (x) = - (12 (2 x - 2) (- 2 u^{- 3} \frac{d}{d x} (x^{2} - 2 x - 3)) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.4.3

Replace all occurrences of $u$ with $x^{2} - 2 x - 3$ .

$f'' (x) = - (12 (2 x - 2) (- 2 {(x^{2} - 2 x - 3)}^{- 3} \frac{d}{d x} (x^{2} - 2 x - 3)) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5

Differentiate.

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Step 3.5.1

Multiply $- 2$ by $12$ .

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} \frac{d}{d x} (x^{2} - 2 x - 3)) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5.2

By the Sum Rule, the derivative of $x^{2} - 2 x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3]$ .

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (- 3))) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5.3

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

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} (2 x + \frac{d}{d x} (- 2 x) + \frac{d}{d x} (- 3))) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5.4

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

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} (2 x - 2 \frac{d}{d x} x + \frac{d}{d x} (- 3))) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5.5

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

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} (2 x - 2 \cdot 1 + \frac{d}{d x} (- 3))) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5.6

Multiply $- 2$ by $1$ .

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} (2 x - 2 + \frac{d}{d x} (- 3))) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5.7

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} (2 x - 2 + 0)) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.5.8

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

$f'' (x) = - (- 24 (2 x - 2) ({(x^{2} - 2 x - 3)}^{- 3} (2 x - 2)) + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.6

Raise $2 x - 2$ to the power of $1$ .

$f'' (x) = - (- 24 ((2 x - 2) (2 x - 2)) {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.7

Raise $2 x - 2$ to the power of $1$ .

$f'' (x) = - (- 24 ((2 x - 2) (2 x - 2)) {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.8

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

$f'' (x) = - (- 24 {(2 x - 2)}^{1 + 1} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.9

Add $1$ and $1$ .

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot \frac{d}{d x} (2 x - 2))$

Step 3.10

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

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot (\frac{d}{d x} (2 x) + \frac{d}{d x} (- 2)))$

Step 3.11

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

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot (2 \frac{d}{d x} (x) + \frac{d}{d x} (- 2)))$

Step 3.12

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

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot (2 \cdot 1 + \frac{d}{d x} (- 2)))$

Step 3.13

Multiply $2$ by $1$ .

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot (2 + \frac{d}{d x} (- 2)))$

Step 3.14

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot (2 + 0))$

Step 3.15

Combine fractions.

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Step 3.15.1

Add $2$ and $0$ .

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12}{{(x^{2} - 2 x - 3)}^{2}} \cdot 2)$

Step 3.15.2

Combine $\frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ and $2$ .

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} + \frac{12 \cdot 2}{{(x^{2} - 2 x - 3)}^{2}})$

Step 3.15.3

Multiply $12$ by $2$ .

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

Step 3.16

To write $- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} - 2 x - 3)}^{2}}{{(x^{2} - 2 x - 3)}^{2}}$ .

$f'' (x) = - (- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3} \cdot \frac{{(x^{2} - 2 x - 3)}^{2}}{{(x^{2} - 2 x - 3)}^{2}} + \frac{24}{{(x^{2} - 2 x - 3)}^{2}})$

Step 3.17

Combine $- 24 {(2 x - 2)}^{2} {(x^{2} - 2 x - 3)}^{- 3}$ and $\frac{{(x^{2} - 2 x - 3)}^{2}}{{(x^{2} - 2 x - 3)}^{2}}$ .

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

Step 3.18

Combine the numerators over the common denominator.

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

Step 3.19

Multiply ${(x^{2} - 2 x - 3)}^{- 3}$ by ${(x^{2} - 2 x - 3)}^{2}$ by adding the exponents.

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Step 3.19.1

Move ${(x^{2} - 2 x - 3)}^{2}$ .

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

Step 3.19.2

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

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

Step 3.19.3

Subtract $3$ from $2$ .

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

Step 3.20

Simplify.

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Step 3.20.1

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

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

Step 3.20.2

Simplify the numerator.

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Step 3.20.2.1

Simplify each term.

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Step 3.20.2.1.1

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

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

Step 3.20.2.1.2

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

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Step 3.20.2.1.2.1

Apply the distributive property.

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

Step 3.20.2.1.2.2

Apply the distributive property.

$f'' (x) = - \frac{- 24 (2 x (2 x) + 2 x \cdot - 2 - 2 (2 x - 2)) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.2.3

Apply the distributive property.

$f'' (x) = - \frac{- 24 (2 x (2 x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3

Simplify and combine like terms.

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Step 3.20.2.1.3.1

Simplify each term.

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Step 3.20.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{- 24 (2 \cdot (2 x \cdot x) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3.1.2

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

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Step 3.20.2.1.3.1.2.1

Move $x$ .

$f'' (x) = - \frac{- 24 (2 \cdot (2 (x \cdot x)) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{- 24 (2 \cdot (2 x^{2}) + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3.1.3

Multiply $2$ by $2$ .

$f'' (x) = - \frac{- 24 (4 x^{2} + 2 x \cdot - 2 - 2 (2 x) - 2 \cdot - 2) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3.1.4

Multiply $- 2$ by $2$ .

$f'' (x) = - \frac{- 24 (4 x^{2} - 4 x - 2 (2 x) - 2 \cdot - 2) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3.1.5

Multiply $2$ by $- 2$ .

$f'' (x) = - \frac{- 24 (4 x^{2} - 4 x - 4 x - 2 \cdot - 2) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3.1.6

Multiply $- 2$ by $- 2$ .

$f'' (x) = - \frac{- 24 (4 x^{2} - 4 x - 4 x + 4) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.3.2

Subtract $4 x$ from $- 4 x$ .

$f'' (x) = - \frac{- 24 (4 x^{2} - 8 x + 4) \frac{1}{x^{2} - 2 x - 3} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.4

Apply the distributive property.

$f'' (x) = - \frac{(- 24 (4 x^{2}) - 24 (- 8 x) - 24 \cdot 4) (\frac{1}{x^{2} - 2 x - 3}) + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.5

Simplify.

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Step 3.20.2.1.5.1

Multiply $4$ by $- 24$ .

$f'' (x) = - \frac{(- 96 x^{2} - 24 (- 8 x) - 24 \cdot 4) (\frac{1}{x^{2} - 2 x - 3}) + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.5.2

Multiply $- 8$ by $- 24$ .

$f'' (x) = - \frac{(- 96 x^{2} + 192 x - 24 \cdot 4) (\frac{1}{x^{2} - 2 x - 3}) + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.5.3

Multiply $- 24$ by $4$ .

$f'' (x) = - \frac{(- 96 x^{2} + 192 x - 96) (\frac{1}{x^{2} - 2 x - 3}) + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.6

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 3.20.2.1.6.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 3.20.2.1.6.2

Write the factored form using these integers.

$f'' (x) = - \frac{(- 96 x^{2} + 192 x - 96) (\frac{1}{(x - 3) (x + 1)}) + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.7

Multiply $- 96 x^{2} + 192 x - 96$ by $\frac{1}{(x - 3) (x + 1)}$ .

$f'' (x) = - \frac{\frac{- 96 x^{2} + 192 x - 96}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8

Simplify the numerator.

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Step 3.20.2.1.8.1

Factor $96$ out of $- 96 x^{2} + 192 x - 96$ .

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Step 3.20.2.1.8.1.1

Factor $96$ out of $- 96 x^{2}$ .

$f'' (x) = - \frac{\frac{96 (- x^{2}) + 192 x - 96}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.1.2

Factor $96$ out of $192 x$ .

$f'' (x) = - \frac{\frac{96 (- x^{2}) + 96 (2 x) - 96}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.1.3

Factor $96$ out of $- 96$ .

$f'' (x) = - \frac{\frac{96 (- x^{2}) + 96 (2 x) + 96 (- 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.1.4

Factor $96$ out of $96 (- x^{2}) + 96 (2 x)$ .

$f'' (x) = - \frac{\frac{96 (- x^{2} + 2 x) + 96 (- 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.1.5

Factor $96$ out of $96 (- x^{2} + 2 x) + 96 (- 1)$ .

$f'' (x) = - \frac{\frac{96 (- x^{2} + 2 x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2

Factor by grouping.

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Step 3.20.2.1.8.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 3.20.2.1.8.2.1.1

Factor $2$ out of $2 x$ .

$f'' (x) = - \frac{\frac{96 (- x^{2} + 2 (x) - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2.1.2

Rewrite $2$ as $1$ plus $1$

$f'' (x) = - \frac{\frac{96 (- x^{2} + (1 + 1) x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2.1.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{96 (- x^{2} + 1 x + 1 x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2.1.4

Multiply $x$ by $1$ .

$f'' (x) = - \frac{\frac{96 (- x^{2} + x + 1 x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2.1.5

Multiply $x$ by $1$ .

$f'' (x) = - \frac{\frac{96 (- x^{2} + x + x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2.2

Factor out the greatest common factor from each group.

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Step 3.20.2.1.8.2.2.1

Group the first two terms and the last two terms.

$f'' (x) = - \frac{\frac{96 ((- x^{2} + x) + x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2.2.2

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = - \frac{\frac{96 (x (- x + 1) - 1 (- x + 1))}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$f'' (x) = - \frac{\frac{96 ((- x + 1) (x - 1))}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

$f'' (x) = - \frac{\frac{96 (- x + 1) (x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3

Combine exponents.

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Step 3.20.2.1.8.3.1

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

$f'' (x) = - \frac{\frac{96 (- (x) + 1) (x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.2

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

$f'' (x) = - \frac{\frac{96 (- (x) - 1 \cdot - 1) (x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.3

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

$f'' (x) = - \frac{\frac{96 (- (x - 1)) (x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.4

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

$f'' (x) = - \frac{\frac{96 (- 1 (x - 1)) (x - 1)}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.5

Raise $x - 1$ to the power of $1$ .

$f'' (x) = - \frac{\frac{96 \cdot (- 1 ((x - 1) (x - 1)))}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.6

Raise $x - 1$ to the power of $1$ .

$f'' (x) = - \frac{\frac{96 \cdot (- 1 ((x - 1) (x - 1)))}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.7

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

$f'' (x) = - \frac{\frac{96 \cdot (- 1 {(x - 1)}^{1 + 1})}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.8

Add $1$ and $1$ .

$f'' (x) = - \frac{\frac{96 \cdot (- 1 {(x - 1)}^{2})}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.8.3.9

Multiply $96$ by $- 1$ .

$f'' (x) = - \frac{\frac{- 96 {(x - 1)}^{2}}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.1.9

Move the negative in front of the fraction.

$f'' (x) = - \frac{- \frac{96 {(x - 1)}^{2}}{(x - 3) (x + 1)} + 24}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.2

To write $24$ as a fraction with a common denominator, multiply by $\frac{(x - 3) (x + 1)}{(x - 3) (x + 1)}$ .

$f'' (x) = - \frac{- \frac{96 {(x - 1)}^{2}}{(x - 3) (x + 1)} + 24 \cdot \frac{(x - 3) (x + 1)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.3

Combine $24$ and $\frac{(x - 3) (x + 1)}{(x - 3) (x + 1)}$ .

$f'' (x) = - \frac{- \frac{96 {(x - 1)}^{2}}{(x - 3) (x + 1)} + \frac{24 ((x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.4

Combine the numerators over the common denominator.

$f'' (x) = - \frac{\frac{- 96 {(x - 1)}^{2} + 24 ((x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5

Simplify the numerator.

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Step 3.20.2.5.1

Factor $24$ out of $- 96 {(x - 1)}^{2} + 24 (x - 3) (x + 1)$ .

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Step 3.20.2.5.1.1

Factor $24$ out of $- 96 {(x - 1)}^{2}$ .

$f'' (x) = - \frac{\frac{24 (- 4 {(x - 1)}^{2}) + 24 (x - 3) (x + 1)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.1.2

Factor $24$ out of $24 (x - 3) (x + 1)$ .

$f'' (x) = - \frac{\frac{24 (- 4 {(x - 1)}^{2}) + 24 ((x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.1.3

Factor $24$ out of $24 (- 4 {(x - 1)}^{2}) + 24 ((x - 3) (x + 1))$ .

$f'' (x) = - \frac{\frac{24 (- 4 {(x - 1)}^{2} + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.2

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

$f'' (x) = - \frac{\frac{24 (- 4 ((x - 1) (x - 1)) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.3

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 3.20.2.5.3.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{24 (- 4 (x (x - 1) - 1 (x - 1)) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.3.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{24 (- 4 (x \cdot x + x \cdot - 1 - 1 (x - 1)) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.3.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{24 (- 4 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.4

Simplify and combine like terms.

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Step 3.20.2.5.4.1

Simplify each term.

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Step 3.20.2.5.4.1.1

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{24 (- 4 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.4.1.2

Move $- 1$ to the left of $x$ .

$f'' (x) = - \frac{\frac{24 (- 4 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.4.1.3

Rewrite $- 1 x$ as $- x$ .

$f'' (x) = - \frac{\frac{24 (- 4 (x^{2} - x - 1 x - 1 \cdot - 1) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.4.1.4

Rewrite $- 1 x$ as $- x$ .

$f'' (x) = - \frac{\frac{24 (- 4 (x^{2} - x - x - 1 \cdot - 1) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.4.1.5

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{\frac{24 (- 4 (x^{2} - x - x + 1) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.4.2

Subtract $x$ from $- x$ .

$f'' (x) = - \frac{\frac{24 (- 4 (x^{2} - 2 x + 1) + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.5

Apply the distributive property.

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} - 4 (- 2 x) - 4 \cdot 1 + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.6

Simplify.

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Step 3.20.2.5.6.1

Multiply $- 2$ by $- 4$ .

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 \cdot 1 + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.6.2

Multiply $- 4$ by $1$ .

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + (x - 3) (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.7

Expand $(x - 3) (x + 1)$ using the FOIL Method.

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Step 3.20.2.5.7.1

Apply the distributive property.

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + x (x + 1) - 3 (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.7.2

Apply the distributive property.

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + x \cdot x + x \cdot 1 - 3 (x + 1))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.7.3

Apply the distributive property.

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + x \cdot x + x \cdot 1 - 3 x - 3 \cdot 1)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.8

Simplify and combine like terms.

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Step 3.20.2.5.8.1

Simplify each term.

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Step 3.20.2.5.8.1.1

Multiply $x$ by $x$ .

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + x^{2} + x \cdot 1 - 3 x - 3 \cdot 1)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.8.1.2

Multiply $x$ by $1$ .

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + x^{2} + x - 3 x - 3 \cdot 1)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.8.1.3

Multiply $- 3$ by $1$ .

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + x^{2} + x - 3 x - 3)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.8.2

Subtract $3 x$ from $x$ .

$f'' (x) = - \frac{\frac{24 (- 4 x^{2} + 8 x - 4 + x^{2} - 2 x - 3)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.9

Add $- 4 x^{2}$ and $x^{2}$ .

$f'' (x) = - \frac{\frac{24 (- 3 x^{2} + 8 x - 4 - 2 x - 3)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.10

Subtract $2 x$ from $8 x$ .

$f'' (x) = - \frac{\frac{24 (- 3 x^{2} + 6 x - 4 - 3)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.5.11

Subtract $3$ from $- 4$ .

$f'' (x) = - \frac{\frac{24 (- 3 x^{2} + 6 x - 7)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.6

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

$f'' (x) = - \frac{\frac{24 (- (3 x^{2}) + 6 x - 7)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.7

Factor $- 1$ out of $6 x$ .

$f'' (x) = - \frac{\frac{24 (- (3 x^{2}) - (- 6 x) - 7)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.8

Factor $- 1$ out of $- (3 x^{2}) - (- 6 x)$ .

$f'' (x) = - \frac{\frac{24 (- (3 x^{2} - 6 x) - 7)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.9

Rewrite $- 7$ as $- 1 (7)$ .

$f'' (x) = - \frac{\frac{24 (- (3 x^{2} - 6 x) - 1 \cdot 7)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.10

Factor $- 1$ out of $- (3 x^{2} - 6 x) - 1 (7)$ .

$f'' (x) = - \frac{\frac{24 (- (3 x^{2} - 6 x + 7))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.11

Rewrite $- (3 x^{2} - 6 x + 7)$ as $- 1 (3 x^{2} - 6 x + 7)$ .

$f'' (x) = - \frac{\frac{24 (- 1 (3 x^{2} - 6 x + 7))}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.2.12

Move the negative in front of the fraction.

$f'' (x) = - \frac{- \frac{24 (3 x^{2} - 6 x + 7)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$

Step 3.20.3

Combine terms.

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Step 3.20.3.1

Rewrite $\frac{- \frac{24 (3 x^{2} - 6 x + 7)}{(x - 3) (x + 1)}}{{(x^{2} - 2 x - 3)}^{2}}$ as a product.

$f'' (x) = - (- \frac{24 (3 x^{2} - 6 x + 7)}{(x - 3) (x + 1)} \cdot \frac{1}{{(x^{2} - 2 x - 3)}^{2}})$

Step 3.20.3.2

Multiply $\frac{1}{{(x^{2} - 2 x - 3)}^{2}}$ by $\frac{24 (3 x^{2} - 6 x + 7)}{(x - 3) (x + 1)}$ .

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x^{2} - 2 x - 3)}^{2} ((x - 3) (x + 1))}$

Step 3.20.3.3

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{24 (3 x^{2} - 6 x + 7)}{{(x^{2} - 2 x - 3)}^{2} (x - 3) (x + 1)})$

Step 3.20.3.4

Multiply $\frac{24 (3 x^{2} - 6 x + 7)}{{(x^{2} - 2 x - 3)}^{2} (x - 3) (x + 1)}$ by $1$ .

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x^{2} - 2 x - 3)}^{2} (x - 3) (x + 1)}$

Step 3.20.4

Reorder terms.

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x^{2} - 2 x - 3)}^{2} (x + 1) (x - 3)}$

Step 3.20.5

Simplify the denominator.

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Step 3.20.5.1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 3.20.5.1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 3.20.5.1.2

Write the factored form using these integers.

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{((x - 3) (x + 1))}^{2} (x + 1) (x - 3)}$

Step 3.20.5.2

Apply the product rule to $(x - 3) (x + 1)$ .

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x - 3)}^{2} {(x + 1)}^{2} (x + 1) (x - 3)}$

Step 3.20.5.3

Combine exponents.

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Step 3.20.5.3.1

Raise $x + 1$ to the power of $1$ .

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x - 3)}^{2} ((x + 1) {(x + 1)}^{2}) (x - 3)}$

Step 3.20.5.3.2

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

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x - 3)}^{2} {(x + 1)}^{1 + 2} (x - 3)}$

Step 3.20.5.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x - 3)}^{2} {(x + 1)}^{3} (x - 3)}$

Step 3.20.5.3.4

Raise $x - 3$ to the power of $1$ .

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{(x - 3) {(x - 3)}^{2} {(x + 1)}^{3}}$

Step 3.20.5.3.5

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

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x - 3)}^{1 + 2} {(x + 1)}^{3}}$

Step 3.20.5.3.6

Add $1$ and $2$ .

$f'' (x) = \frac{24 (3 x^{2} - 6 x + 7)}{{(x - 3)}^{3} {(x + 1)}^{3}}$

Step 4

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

$- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}} = 0$

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 $12$ is constant with respect to $x$ , the derivative of $\frac{12}{x^{2} - 2 x - 3}$ with respect to $x$ is $12 \frac{d}{d x} [\frac{1}{x^{2} - 2 x - 3}]$ .

$12 \frac{d}{d x} [\frac{1}{x^{2} - 2 x - 3}]$

Step 5.1.1.2

Rewrite $\frac{1}{x^{2} - 2 x - 3}$ as ${(x^{2} - 2 x - 3)}^{- 1}$ .

$12 \frac{d}{d x} [{(x^{2} - 2 x - 3)}^{- 1}]$

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 $g (x) = x^{2} - 2 x - 3$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 2 x - 3$ .

$12 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} - 2 x - 3])$

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 $n = - 1$ .

$12 (- u^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 5.1.2.3

Replace all occurrences of $u$ with $x^{2} - 2 x - 3$ .

$12 (- {(x^{2} - 2 x - 3)}^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 5.1.3

Differentiate.

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Step 5.1.3.1

Multiply $- 1$ by $12$ .

$- 12 ({(x^{2} - 2 x - 3)}^{- 2} \frac{d}{d x} [x^{2} - 2 x - 3])$

Step 5.1.3.2

By the Sum Rule, the derivative of $x^{2} - 2 x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3]$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3])$

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 $n = 2$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [- 3])$

Step 5.1.3.4

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

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

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 $n = 1$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 \cdot 1 + \frac{d}{d x} [- 3])$

Step 5.1.3.6

Multiply $- 2$ by $1$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 + \frac{d}{d x} [- 3])$

Step 5.1.3.7

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2 + 0)$

Step 5.1.3.8

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

$- 12 {(x^{2} - 2 x - 3)}^{- 2} (2 x - 2)$

Step 5.1.4

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

$- 12 \frac{1}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$

Step 5.1.5

Simplify.

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Step 5.1.5.1

Combine terms.

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Step 5.1.5.1.1

Combine $- 12$ and $\frac{1}{{(x^{2} - 2 x - 3)}^{2}}$ .

$\frac{- 12}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$

Step 5.1.5.1.2

Move the negative in front of the fraction.

$- \frac{12}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$

Step 5.1.5.2

Reorder the factors of $- \frac{12}{{(x^{2} - 2 x - 3)}^{2}} (2 x - 2)$ .

$f' (x) = - (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ .

$- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}} = 0$

Step 6.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 x - 2 = 0$

$\frac{12}{{(x^{2} - 2 x - 3)}^{2}} = 0$

Step 6.3

Set $2 x - 2$ equal to $0$ and solve for $x$ .

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Step 6.3.1

Set $2 x - 2$ equal to $0$ .

$2 x - 2 = 0$

Step 6.3.2

Solve $2 x - 2 = 0$ for $x$ .

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Step 6.3.2.1

Add $2$ to both sides of the equation.

$2 x = 2$

Step 6.3.2.2

Divide each term in $2 x = 2$ by $2$ and simplify.

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Step 6.3.2.2.1

Divide each term in $2 x = 2$ by $2$ .

$\frac{2 x}{2} = \frac{2}{2}$

Step 6.3.2.2.2

Simplify the left side.

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Step 6.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2}{2}$

Step 6.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{2}$

Step 6.3.2.2.3

Simplify the right side.

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Step 6.3.2.2.3.1

Divide $2$ by $2$ .

$x = 1$

Step 6.4

Set $\frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $\frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ equal to $0$ .

$\frac{12}{{(x^{2} - 2 x - 3)}^{2}} = 0$

Step 6.4.2

Solve $\frac{12}{{(x^{2} - 2 x - 3)}^{2}} = 0$ for $x$ .

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Step 6.4.2.1

Set the numerator equal to zero.

$12 = 0$

Step 6.4.2.2

Since $12 \neq 0$ , there are no solutions.

No solution

Step 6.5

The final solution is all the values that make $- (2 x - 2) \frac{12}{{(x^{2} - 2 x - 3)}^{2}} = 0$ true.

$x = 1$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{12}{{(x^{2} - 2 x - 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{2} - 2 x - 3)}^{2} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Factor the left side of the equation.

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Step 7.2.1.1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 7.2.1.1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 7.2.1.1.2

Write the factored form using these integers.

${((x - 3) (x + 1))}^{2} = 0$

Step 7.2.1.2

Apply the product rule to $(x - 3) (x + 1)$ .

${(x - 3)}^{2} {(x + 1)}^{2} = 0$

Step 7.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 3)}^{2} = 0$

${(x + 1)}^{2} = 0$

Step 7.2.3

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 7.2.3.1

Set ${(x - 3)}^{2}$ equal to $0$ .

${(x - 3)}^{2} = 0$

Step 7.2.3.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 7.2.3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 7.2.3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7.2.4

Set ${(x + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 7.2.4.1

Set ${(x + 1)}^{2}$ equal to $0$ .

${(x + 1)}^{2} = 0$

Step 7.2.4.2

Solve ${(x + 1)}^{2} = 0$ for $x$ .

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Step 7.2.4.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 7.2.4.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7.2.5

The final solution is all the values that make ${(x - 3)}^{2} {(x + 1)}^{2} = 0$ true.

$x = 3, - 1$

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 $0$ .

$x = - 1, x = 3$

Step 8

Critical points to evaluate.

$x = 1$

Step 9

Evaluate the second derivative at $x = 1$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{24 (3 {(1)}^{2} - 6 \cdot 1 + 7)}{{((1) - 3)}^{3} {((1) + 1)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

One to any power is one.

$\frac{24 (3 \cdot 1 - 6 \cdot 1 + 7)}{{(1 - 3)}^{3} {(1 + 1)}^{3}}$

Step 10.1.2

Multiply $3$ by $1$ .

$\frac{24 (3 - 6 \cdot 1 + 7)}{{(1 - 3)}^{3} {(1 + 1)}^{3}}$

Step 10.1.3

Multiply $- 6$ by $1$ .

$\frac{24 (3 - 6 + 7)}{{(1 - 3)}^{3} {(1 + 1)}^{3}}$

Step 10.1.4

Subtract $6$ from $3$ .

$\frac{24 (- 3 + 7)}{{(1 - 3)}^{3} {(1 + 1)}^{3}}$

Step 10.1.5

Add $- 3$ and $7$ .

$\frac{24 \cdot 4}{{(1 - 3)}^{3} {(1 + 1)}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Subtract $3$ from $1$ .

$\frac{24 \cdot 4}{{(- 2)}^{3} {(1 + 1)}^{3}}$

Step 10.2.2

Add $1$ and $1$ .

$\frac{24 \cdot 4}{{(- 2)}^{3} \cdot 2^{3}}$

Step 10.2.3

Raise $- 2$ to the power of $3$ .

$\frac{24 \cdot 4}{- 8 \cdot 2^{3}}$

Step 10.2.4

Raise $2$ to the power of $3$ .

$\frac{24 \cdot 4}{- 8 \cdot 8}$

Step 10.3

Reduce the expression by cancelling the common factors.

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Step 10.3.1

Multiply $24$ by $4$ .

$\frac{96}{- 8 \cdot 8}$

Step 10.3.2

Multiply $- 8$ by $8$ .

$\frac{96}{- 64}$

Step 10.3.3

Cancel the common factor of $96$ and $- 64$ .

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Step 10.3.3.1

Factor $32$ out of $96$ .

$\frac{32 (3)}{- 64}$

Step 10.3.3.2

Cancel the common factors.

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Step 10.3.3.2.1

Factor $32$ out of $- 64$ .

$\frac{32 \cdot 3}{32 \cdot - 2}$

Step 10.3.3.2.2

Cancel the common factor.

$\frac{32 \cdot 3}{32 \cdot - 2}$

Step 10.3.3.2.3

Rewrite the expression.

$\frac{3}{- 2}$

Step 10.3.4

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 11

$x = 1$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1$ is a local maximum

Step 12

Find the y-value when $x = 1$ .

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Step 12.1

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

$f (1) = \frac{12}{{(1)}^{2} - 2 \cdot 1 - 3}$

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

One to any power is one.

$f (1) = \frac{12}{1 - 2 \cdot 1 - 3}$

Step 12.2.1.2

Multiply $- 2$ by $1$ .

$f (1) = \frac{12}{1 - 2 - 3}$

Step 12.2.1.3

Subtract $2$ from $1$ .

$f (1) = \frac{12}{- 1 - 3}$

Step 12.2.1.4

Subtract $3$ from $- 1$ .

$f (1) = \frac{12}{- 4}$

Step 12.2.2

Divide $12$ by $- 4$ .

$f (1) = - 3$

Step 12.2.3

The final answer is $- 3$ .

$y = - 3$

Step 13

These are the local extrema for $f (x) = \frac{12}{x^{2} - 2 x - 3}$ .

$(1, - 3)$ is a local maxima

Step 14