Calculus Examples

$f (x) = \frac{x^{2} - x - 2}{x^{2} - 6 x + 9}$

Step 1

Find the first derivative of the function.

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Step 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 $f (x) = x^{2} - x - 2$ and $g (x) = x^{2} - 6 x + 9$ .

$\frac{(x^{2} - 6 x + 9) \frac{d}{d x} [x^{2} - x - 2] - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{(x^{2} - 6 x + 9) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.2

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

$\frac{(x^{2} - 6 x + 9) (2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.3

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

$\frac{(x^{2} - 6 x + 9) (2 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.4

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1 \cdot 1 + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.5

Multiply $- 1$ by $1$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1 + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.6

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1 + 0) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.7

Add $2 x - 1$ and $0$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.8

By the Sum Rule, the derivative of $x^{2} - 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.9

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.10

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.11

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 \cdot 1 + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.12

Multiply $- 6$ by $1$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.13

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 + 0)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.2.14

Add $2 x - 6$ and $0$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{(x^{2} - 6 x + 9) (2 x - 1) + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2

Simplify the numerator.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Expand $(x^{2} - 6 x + 9) (2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} (2 x) + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2

Simplify each term.

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Step 1.3.2.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} x + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.2.1.2.2.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.3.2.1.2.2.2.1

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

$\frac{2 (x^{1} x^{2}) + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.2.2.2

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

$\frac{2 x^{1 + 2} + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.3

Move $- 1$ to the left of $x^{2}$ .

$\frac{2 x^{3} - 1 \cdot x^{2} - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.4

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

$\frac{2 x^{3} - x^{2} - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - x^{2} - 6 \cdot 2 x \cdot x - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.6

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

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Step 1.3.2.1.2.6.1

Move $x$ .

$\frac{2 x^{3} - x^{2} - 6 \cdot 2 (x \cdot x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - x^{2} - 6 \cdot 2 x^{2} - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.7

Multiply $- 6$ by $2$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.8

Multiply $- 1$ by $- 6$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} + 6 x + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.9

Multiply $2$ by $9$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} + 6 x + 18 x + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.2.10

Multiply $9$ by $- 1$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} + 6 x + 18 x - 9 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.3

Subtract $12 x^{2}$ from $- x^{2}$ .

$\frac{2 x^{3} - 13 x^{2} + 6 x + 18 x - 9 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.4

Add $6 x$ and $18 x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.5

Simplify each term.

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Step 1.3.2.1.5.1

Multiply $- - x$ .

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Step 1.3.2.1.5.1.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} + 1 x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.5.1.2

Multiply $x$ by $1$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} + x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.5.2

Multiply $- 1$ by $- 2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} + x + 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.6

Expand $(- x^{2} + x + 2) (2 x - 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - x^{2} (2 x) - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7

Simplify each term.

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Step 1.3.2.1.7.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 x^{2} x - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.3.2.1.7.2.1

Move $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.2.2

Multiply $x$ by $x^{2}$ .

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Step 1.3.2.1.7.2.2.1

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

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.2.2.2

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

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 x^{3} - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.4

Multiply $- 6$ by $- 1$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x \cdot x + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.6

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

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Step 1.3.2.1.7.6.1

Move $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 (x \cdot x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.7

Move $- 6$ to the left of $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} - 6 \cdot x + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.8

Multiply $2$ by $2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} - 6 x + 4 x + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.7.9

Multiply $2$ by $- 6$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} - 6 x + 4 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.8

Add $6 x^{2}$ and $2 x^{2}$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 8 x^{2} - 6 x + 4 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.1.9

Add $- 6 x$ and $4 x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 8 x^{2} - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.2

Combine the opposite terms in $2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 8 x^{2} - 2 x - 12$ .

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Step 1.3.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{- 13 x^{2} + 24 x - 9 + 0 + 8 x^{2} - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.2.2

Add $- 13 x^{2} + 24 x - 9$ and $0$ .

$\frac{- 13 x^{2} + 24 x - 9 + 8 x^{2} - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.3

Add $- 13 x^{2}$ and $8 x^{2}$ .

$\frac{- 5 x^{2} + 24 x - 9 - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.4

Subtract $2 x$ from $24 x$ .

$\frac{- 5 x^{2} + 22 x - 9 - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.2.5

Subtract $12$ from $- 9$ .

$\frac{- 5 x^{2} + 22 x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.3

Factor by grouping.

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Step 1.3.3.1

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 $a \cdot c = - 5 \cdot - 21 = 105$ and whose sum is $b = 22$ .

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Step 1.3.3.1.1

Factor $22$ out of $22 x$ .

$\frac{- 5 x^{2} + 22 (x) - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.3.1.2

Rewrite $22$ as $7$ plus $15$

$\frac{- 5 x^{2} + (7 + 15) x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.3.1.3

Apply the distributive property.

$\frac{- 5 x^{2} + 7 x + 15 x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.3.2

Factor out the greatest common factor from each group.

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Step 1.3.3.2.1

Group the first two terms and the last two terms.

$\frac{(- 5 x^{2} + 7 x) + 15 x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.3.2.2

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

$\frac{x (- 5 x + 7) - 3 (- 5 x + 7)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.3.3

Factor the polynomial by factoring out the greatest common factor, $- 5 x + 7$ .

$\frac{(- 5 x + 7) (x - 3)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 1.3.4

Simplify the denominator.

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Step 1.3.4.1

Factor using the perfect square rule.

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Step 1.3.4.1.1

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

$\frac{(- 5 x + 7) (x - 3)}{{(x^{2} - 6 x + 3^{2})}^{2}}$

Step 1.3.4.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.3.4.1.3

Rewrite the polynomial.

$\frac{(- 5 x + 7) (x - 3)}{{(x^{2} - 2 \cdot x \cdot 3 + 3^{2})}^{2}}$

Step 1.3.4.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{(- 5 x + 7) (x - 3)}{{({(x - 3)}^{2})}^{2}}$

Step 1.3.4.2

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

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Step 1.3.4.2.1

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

$\frac{(- 5 x + 7) (x - 3)}{{(x - 3)}^{2 \cdot 2}}$

Step 1.3.4.2.2

Multiply $2$ by $2$ .

$\frac{(- 5 x + 7) (x - 3)}{{(x - 3)}^{4}}$

Step 1.3.4.3

Use the Binomial Theorem.

$\frac{(- 5 x + 7) (x - 3)}{x^{4} + 4 x^{3} \cdot - 3 + 6 x^{2} {(- 3)}^{2} + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 1.3.4.4

Simplify each term.

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Step 1.3.4.4.1

Multiply $- 3$ by $4$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 6 x^{2} {(- 3)}^{2} + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 1.3.4.4.2

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

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 6 x^{2} \cdot 9 + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 1.3.4.4.3

Multiply $9$ by $6$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 1.3.4.4.4

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

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} + 4 x \cdot - 27 + {(- 3)}^{4}}$

Step 1.3.4.4.5

Multiply $- 27$ by $4$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + {(- 3)}^{4}}$

Step 1.3.4.4.6

Raise $- 3$ to the power of $4$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81}$

Step 1.3.4.5

Make each term match the terms from the binomial theorem formula.

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 4 \cdot 3 x^{3} + 6 \cdot 3^{2} x^{2} - 4 \cdot 3^{3} x + 3^{4}}$

Step 1.3.4.6

Factor $x^{4} - 4 \cdot 3 x^{3} + 6 \cdot 3^{2} x^{2} - 4 \cdot 3^{3} x + 3^{4}$ using the binomial theorem.

$\frac{(- 5 x + 7) (x - 3)}{{(3 - x)}^{4}}$

Step 1.3.5

Cancel the common factor of $x - 3$ and ${(3 - x)}^{4}$ .

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Step 1.3.5.1

Factor $- 1$ out of $x$ .

$\frac{(- 5 x + 7) (- 1 (- x) - 3)}{{(3 - x)}^{4}}$

Step 1.3.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{(- 5 x + 7) (- 1 (- x) - 1 (3))}{{(3 - x)}^{4}}$

Step 1.3.5.3

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

$\frac{(- 5 x + 7) (- 1 (- x + 3))}{{(3 - x)}^{4}}$

Step 1.3.5.4

Reorder terms.

$\frac{(- 5 x + 7) (- 1 (- x + 3))}{{(- x + 3)}^{4}}$

Step 1.3.5.5

Factor $- x + 3$ out of $(- 5 x + 7) (- 1 (- x + 3))$ .

$\frac{(- x + 3) ((- 5 x + 7) \cdot (- 1))}{{(- x + 3)}^{4}}$

Step 1.3.5.6

Cancel the common factors.

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Step 1.3.5.6.1

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

$\frac{(- x + 3) ((- 5 x + 7) \cdot (- 1))}{(- x + 3) {(- x + 3)}^{3}}$

Step 1.3.5.6.2

Cancel the common factor.

$\frac{(- x + 3) ((- 5 x + 7) \cdot (- 1))}{(- x + 3) {(- x + 3)}^{3}}$

Step 1.3.5.6.3

Rewrite the expression.

$\frac{(- 5 x + 7) \cdot (- 1)}{{(- x + 3)}^{3}}$

Step 1.3.6

Move $- 1$ to the left of $- 5 x + 7$ .

$\frac{- 1 \cdot (- 5 x + 7)}{{(- x + 3)}^{3}}$

Step 1.3.7

Move the negative in front of the fraction.

$- \frac{- 5 x + 7}{{(- x + 3)}^{3}}$

Step 1.3.8

Factor $- 1$ out of $- 5 x$ .

$- \frac{- (5 x) + 7}{{(- x + 3)}^{3}}$

Step 1.3.9

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

$- \frac{- (5 x) - 1 (- 7)}{{(- x + 3)}^{3}}$

Step 1.3.10

Factor $- 1$ out of $- (5 x) - 1 (- 7)$ .

$- \frac{- (5 x - 7)}{{(- x + 3)}^{3}}$

Step 1.3.11

Rewrite $- (5 x - 7)$ as $- 1 (5 x - 7)$ .

$- \frac{- 1 (5 x - 7)}{{(- x + 3)}^{3}}$

Step 1.3.12

Move the negative in front of the fraction.

$- - \frac{5 x - 7}{{(- x + 3)}^{3}}$

Step 1.3.13

Multiply $- 1$ by $- 1$ .

$1 \frac{5 x - 7}{{(- x + 3)}^{3}}$

Step 1.3.14

Multiply $\frac{5 x - 7}{{(- x + 3)}^{3}}$ by $1$ .

$\frac{5 x - 7}{{(- x + 3)}^{3}}$

Step 2

Find the second derivative of the function.

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Step 2.1

$f'' (x) = \frac{{(- x + 3)}^{3} \frac{d}{d x} (5 x - 7) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{({(- x + 3)}^{3})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

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

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Step 2.2.1.1

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

$f'' (x) = \frac{{(- x + 3)}^{3} \frac{d}{d x} (5 x - 7) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{3 \cdot 2}}$

Step 2.2.1.2

Multiply $3$ by $2$ .

$f'' (x) = \frac{{(- x + 3)}^{3} \frac{d}{d x} (5 x - 7) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.2.2

By the Sum Rule, the derivative of $5 x - 7$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 7]$ .

$f'' (x) = \frac{{(- x + 3)}^{3} (\frac{d}{d x} (5 x) + \frac{d}{d x} (- 7)) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.2.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{{(- x + 3)}^{3} (5 \frac{d}{d x} (x) + \frac{d}{d x} (- 7)) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.2.4

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

$f'' (x) = \frac{{(- x + 3)}^{3} (5 \cdot 1 + \frac{d}{d x} (- 7)) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.2.5

Multiply $5$ by $1$ .

$f'' (x) = \frac{{(- x + 3)}^{3} (5 + \frac{d}{d x} (- 7)) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.2.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(- x + 3)}^{3} (5 + 0) - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.2.7

Simplify the expression.

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Step 2.2.7.1

Add $5$ and $0$ .

$f'' (x) = \frac{{(- x + 3)}^{3} \cdot 5 - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.2.7.2

Move $5$ to the left of ${(- x + 3)}^{3}$ .

$f'' (x) = \frac{5 \cdot {(- x + 3)}^{3} - (5 x - 7) \frac{d}{d x} {(- x + 3)}^{3}}{{(- x + 3)}^{6}}$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = - x + 3$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $- x + 3$ .

$f'' (x) = \frac{5 {(- x + 3)}^{3} - (5 x - 7) (\frac{d}{d u} (u^{3}) \frac{d}{d x} (- x + 3))}{{(- x + 3)}^{6}}$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$f'' (x) = \frac{5 {(- x + 3)}^{3} - (5 x - 7) (3 u^{2} \frac{d}{d x} (- x + 3))}{{(- x + 3)}^{6}}$

Step 2.3.3

Replace all occurrences of $u$ with $- x + 3$ .

$f'' (x) = \frac{5 {(- x + 3)}^{3} - (5 x - 7) (3 {(- x + 3)}^{2} \frac{d}{d x} (- x + 3))}{{(- x + 3)}^{6}}$

Step 2.4

Simplify with factoring out.

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Step 2.4.1

Multiply $3$ by $- 1$ .

$f'' (x) = \frac{5 {(- x + 3)}^{3} - 3 (5 x - 7) ({(- x + 3)}^{2} \frac{d}{d x} (- x + 3))}{{(- x + 3)}^{6}}$

Step 2.4.2

Factor ${(- x + 3)}^{2}$ out of $5 {(- x + 3)}^{3} - 3 (5 x - 7) ({(- x + 3)}^{2} \frac{d}{d x} [- x + 3])$ .

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Step 2.4.2.1

Factor ${(- x + 3)}^{2}$ out of $5 {(- x + 3)}^{3}$ .

$f'' (x) = \frac{{(- x + 3)}^{2} (5 (- x + 3)) - 3 (5 x - 7) ({(- x + 3)}^{2} \frac{d}{d x} (- x + 3))}{{(- x + 3)}^{6}}$

Step 2.4.2.2

Factor ${(- x + 3)}^{2}$ out of $- 3 (5 x - 7) ({(- x + 3)}^{2} \frac{d}{d x} [- x + 3])$ .

$f'' (x) = \frac{{(- x + 3)}^{2} (5 (- x + 3)) + {(- x + 3)}^{2} (- 3 (5 x - 7) (\frac{d}{d x} (- x + 3)))}{{(- x + 3)}^{6}}$

Step 2.4.2.3

Factor ${(- x + 3)}^{2}$ out of ${(- x + 3)}^{2} (5 (- x + 3)) + {(- x + 3)}^{2} (- 3 (5 x - 7) (\frac{d}{d x} [- x + 3]))$ .

$f'' (x) = \frac{{(- x + 3)}^{2} (5 (- x + 3) - 3 (5 x - 7) (\frac{d}{d x} (- x + 3)))}{{(- x + 3)}^{6}}$

Step 2.5

Cancel the common factors.

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Step 2.5.1

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

$f'' (x) = \frac{{(- x + 3)}^{2} (5 (- x + 3) - 3 (5 x - 7) (\frac{d}{d x} (- x + 3)))}{{(- x + 3)}^{2} {(- x + 3)}^{4}}$

Step 2.5.2

Cancel the common factor.

$f'' (x) = \frac{{(- x + 3)}^{2} (5 (- x + 3) - 3 (5 x - 7) (\frac{d}{d x} (- x + 3)))}{{(- x + 3)}^{2} {(- x + 3)}^{4}}$

Step 2.5.3

Rewrite the expression.

$f'' (x) = \frac{5 (- x + 3) - 3 (5 x - 7) (\frac{d}{d x} (- x + 3))}{{(- x + 3)}^{4}}$

Step 2.6

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

$f'' (x) = \frac{5 (- x + 3) - 3 (5 x - 7) (\frac{d}{d x} (- x) + \frac{d}{d x} (3))}{{(- x + 3)}^{4}}$

Step 2.7

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

$f'' (x) = \frac{5 (- x + 3) - 3 (5 x - 7) (- \frac{d}{d x} x + \frac{d}{d x} (3))}{{(- x + 3)}^{4}}$

Step 2.8

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

$f'' (x) = \frac{5 (- x + 3) - 3 (5 x - 7) (- 1 \cdot 1 + \frac{d}{d x} (3))}{{(- x + 3)}^{4}}$

Step 2.9

Multiply $- 1$ by $1$ .

$f'' (x) = \frac{5 (- x + 3) - 3 (5 x - 7) (- 1 + \frac{d}{d x} (3))}{{(- x + 3)}^{4}}$

Step 2.10

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

$f'' (x) = \frac{5 (- x + 3) - 3 (5 x - 7) (- 1 + 0)}{{(- x + 3)}^{4}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $- 1$ and $0$ .

$f'' (x) = \frac{5 (- x + 3) - 3 (5 x - 7) \cdot - 1}{{(- x + 3)}^{4}}$

Step 2.11.2

Multiply $- 1$ by $- 3$ .

$f'' (x) = \frac{5 (- x + 3) + 3 (5 x - 7)}{{(- x + 3)}^{4}}$

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

$f'' (x) = \frac{5 (- x) + 5 \cdot 3 + 3 (5 x - 7)}{{(- x + 3)}^{4}}$

Step 2.12.2

Apply the distributive property.

$f'' (x) = \frac{5 (- x) + 5 \cdot 3 + 3 (5 x) + 3 \cdot - 7}{{(- x + 3)}^{4}}$

Step 2.12.3

Simplify the numerator.

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Step 2.12.3.1

Simplify each term.

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Step 2.12.3.1.1

Multiply $- 1$ by $5$ .

$f'' (x) = \frac{- 5 x + 5 \cdot 3 + 3 (5 x) + 3 \cdot - 7}{{(- x + 3)}^{4}}$

Step 2.12.3.1.2

Multiply $5$ by $3$ .

$f'' (x) = \frac{- 5 x + 15 + 3 (5 x) + 3 \cdot - 7}{{(- x + 3)}^{4}}$

Step 2.12.3.1.3

Multiply $5$ by $3$ .

$f'' (x) = \frac{- 5 x + 15 + 15 x + 3 \cdot - 7}{{(- x + 3)}^{4}}$

Step 2.12.3.1.4

Multiply $3$ by $- 7$ .

$f'' (x) = \frac{- 5 x + 15 + 15 x - 21}{{(- x + 3)}^{4}}$

Step 2.12.3.2

Add $- 5 x$ and $15 x$ .

$f'' (x) = \frac{10 x + 15 - 21}{{(- x + 3)}^{4}}$

Step 2.12.3.3

Subtract $21$ from $15$ .

$f'' (x) = \frac{10 x - 6}{{(- x + 3)}^{4}}$

Step 2.12.4

Factor $2$ out of $10 x - 6$ .

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Step 2.12.4.1

Factor $2$ out of $10 x$ .

$f'' (x) = \frac{2 (5 x) - 6}{{(- x + 3)}^{4}}$

Step 2.12.4.2

Factor $2$ out of $- 6$ .

$f'' (x) = \frac{2 (5 x) + 2 (- 3)}{{(- x + 3)}^{4}}$

Step 2.12.4.3

Factor $2$ out of $2 (5 x) + 2 (- 3)$ .

$f'' (x) = \frac{2 (5 x - 3)}{{(- x + 3)}^{4}}$

Step 3

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

$\frac{5 x - 7}{{(- x + 3)}^{3}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{(x^{2} - 6 x + 9) \frac{d}{d x} [x^{2} - x - 2] - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

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

$\frac{(x^{2} - 6 x + 9) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.2

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

$\frac{(x^{2} - 6 x + 9) (2 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.3

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

$\frac{(x^{2} - 6 x + 9) (2 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.4

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1 \cdot 1 + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.5

Multiply $- 1$ by $1$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1 + \frac{d}{d x} [- 2]) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.6

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1 + 0) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.7

Add $2 x - 1$ and $0$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) \frac{d}{d x} [x^{2} - 6 x + 9]}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.8

By the Sum Rule, the derivative of $x^{2} - 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.9

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.10

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.11

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

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 \cdot 1 + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.12

Multiply $- 6$ by $1$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 + \frac{d}{d x} [9])}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.13

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6 + 0)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.2.14

Add $2 x - 6$ and $0$ .

$\frac{(x^{2} - 6 x + 9) (2 x - 1) - (x^{2} - x - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Apply the distributive property.

$\frac{(x^{2} - 6 x + 9) (2 x - 1) + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2

Simplify the numerator.

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Step 4.1.3.2.1

Simplify each term.

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Step 4.1.3.2.1.1

Expand $(x^{2} - 6 x + 9) (2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} (2 x) + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2

Simplify each term.

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Step 4.1.3.2.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} x + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.3.2.1.2.2.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.2.2

Multiply $x$ by $x^{2}$ .

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Step 4.1.3.2.1.2.2.2.1

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

$\frac{2 (x^{1} x^{2}) + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.2.2.2

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

$\frac{2 x^{1 + 2} + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot - 1 - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.3

Move $- 1$ to the left of $x^{2}$ .

$\frac{2 x^{3} - 1 \cdot x^{2} - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.4

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

$\frac{2 x^{3} - x^{2} - 6 x (2 x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - x^{2} - 6 \cdot 2 x \cdot x - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.6

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

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Step 4.1.3.2.1.2.6.1

Move $x$ .

$\frac{2 x^{3} - x^{2} - 6 \cdot 2 (x \cdot x) - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - x^{2} - 6 \cdot 2 x^{2} - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.7

Multiply $- 6$ by $2$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} - 6 x \cdot - 1 + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.8

Multiply $- 1$ by $- 6$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} + 6 x + 9 (2 x) + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.9

Multiply $2$ by $9$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} + 6 x + 18 x + 9 \cdot - 1 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.2.10

Multiply $9$ by $- 1$ .

$\frac{2 x^{3} - x^{2} - 12 x^{2} + 6 x + 18 x - 9 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.3

Subtract $12 x^{2}$ from $- x^{2}$ .

$\frac{2 x^{3} - 13 x^{2} + 6 x + 18 x - 9 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.4

Add $6 x$ and $18 x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} - - x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.5

Simplify each term.

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Step 4.1.3.2.1.5.1

Multiply $- - x$ .

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Step 4.1.3.2.1.5.1.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} + 1 x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.5.1.2

Multiply $x$ by $1$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} + x - - 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.5.2

Multiply $- 1$ by $- 2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 + (- x^{2} + x + 2) (2 x - 6)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.6

Expand $(- x^{2} + x + 2) (2 x - 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - x^{2} (2 x) - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7

Simplify each term.

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Step 4.1.3.2.1.7.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 x^{2} x - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.3.2.1.7.2.1

Move $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.2.2

Multiply $x$ by $x^{2}$ .

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Step 4.1.3.2.1.7.2.2.1

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

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.2.2.2

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

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 1 \cdot 2 x^{3} - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} - x^{2} \cdot - 6 + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.4

Multiply $- 6$ by $- 1$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + x (2 x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x \cdot x + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.6

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

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Step 4.1.3.2.1.7.6.1

Move $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 (x \cdot x) + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} + x \cdot - 6 + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.7

Move $- 6$ to the left of $x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} - 6 \cdot x + 2 (2 x) + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.8

Multiply $2$ by $2$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} - 6 x + 4 x + 2 \cdot - 6}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.7.9

Multiply $2$ by $- 6$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 6 x^{2} + 2 x^{2} - 6 x + 4 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.8

Add $6 x^{2}$ and $2 x^{2}$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 8 x^{2} - 6 x + 4 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.1.9

Add $- 6 x$ and $4 x$ .

$\frac{2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 8 x^{2} - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.2

Combine the opposite terms in $2 x^{3} - 13 x^{2} + 24 x - 9 - 2 x^{3} + 8 x^{2} - 2 x - 12$ .

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Step 4.1.3.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{- 13 x^{2} + 24 x - 9 + 0 + 8 x^{2} - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.2.2

Add $- 13 x^{2} + 24 x - 9$ and $0$ .

$\frac{- 13 x^{2} + 24 x - 9 + 8 x^{2} - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.3

Add $- 13 x^{2}$ and $8 x^{2}$ .

$\frac{- 5 x^{2} + 24 x - 9 - 2 x - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.4

Subtract $2 x$ from $24 x$ .

$\frac{- 5 x^{2} + 22 x - 9 - 12}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.2.5

Subtract $12$ from $- 9$ .

$\frac{- 5 x^{2} + 22 x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.3

Factor by grouping.

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Step 4.1.3.3.1

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 $a \cdot c = - 5 \cdot - 21 = 105$ and whose sum is $b = 22$ .

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Step 4.1.3.3.1.1

Factor $22$ out of $22 x$ .

$\frac{- 5 x^{2} + 22 (x) - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.3.1.2

Rewrite $22$ as $7$ plus $15$

$\frac{- 5 x^{2} + (7 + 15) x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.3.1.3

Apply the distributive property.

$\frac{- 5 x^{2} + 7 x + 15 x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.3.2

Factor out the greatest common factor from each group.

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Step 4.1.3.3.2.1

Group the first two terms and the last two terms.

$\frac{(- 5 x^{2} + 7 x) + 15 x - 21}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.3.2.2

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

$\frac{x (- 5 x + 7) - 3 (- 5 x + 7)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.3.3

Factor the polynomial by factoring out the greatest common factor, $- 5 x + 7$ .

$\frac{(- 5 x + 7) (x - 3)}{{(x^{2} - 6 x + 9)}^{2}}$

Step 4.1.3.4

Simplify the denominator.

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Step 4.1.3.4.1

Factor using the perfect square rule.

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Step 4.1.3.4.1.1

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

$\frac{(- 5 x + 7) (x - 3)}{{(x^{2} - 6 x + 3^{2})}^{2}}$

Step 4.1.3.4.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 4.1.3.4.1.3

Rewrite the polynomial.

$\frac{(- 5 x + 7) (x - 3)}{{(x^{2} - 2 \cdot x \cdot 3 + 3^{2})}^{2}}$

Step 4.1.3.4.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{(- 5 x + 7) (x - 3)}{{({(x - 3)}^{2})}^{2}}$

Step 4.1.3.4.2

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

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Step 4.1.3.4.2.1

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

$\frac{(- 5 x + 7) (x - 3)}{{(x - 3)}^{2 \cdot 2}}$

Step 4.1.3.4.2.2

Multiply $2$ by $2$ .

$\frac{(- 5 x + 7) (x - 3)}{{(x - 3)}^{4}}$

Step 4.1.3.4.3

Use the Binomial Theorem.

$\frac{(- 5 x + 7) (x - 3)}{x^{4} + 4 x^{3} \cdot - 3 + 6 x^{2} {(- 3)}^{2} + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 4.1.3.4.4

Simplify each term.

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Step 4.1.3.4.4.1

Multiply $- 3$ by $4$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 6 x^{2} {(- 3)}^{2} + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 4.1.3.4.4.2

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

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 6 x^{2} \cdot 9 + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 4.1.3.4.4.3

Multiply $9$ by $6$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} + 4 x {(- 3)}^{3} + {(- 3)}^{4}}$

Step 4.1.3.4.4.4

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

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} + 4 x \cdot - 27 + {(- 3)}^{4}}$

Step 4.1.3.4.4.5

Multiply $- 27$ by $4$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + {(- 3)}^{4}}$

Step 4.1.3.4.4.6

Raise $- 3$ to the power of $4$ .

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81}$

Step 4.1.3.4.5

Make each term match the terms from the binomial theorem formula.

$\frac{(- 5 x + 7) (x - 3)}{x^{4} - 4 \cdot 3 x^{3} + 6 \cdot 3^{2} x^{2} - 4 \cdot 3^{3} x + 3^{4}}$

Step 4.1.3.4.6

Factor $x^{4} - 4 \cdot 3 x^{3} + 6 \cdot 3^{2} x^{2} - 4 \cdot 3^{3} x + 3^{4}$ using the binomial theorem.

$\frac{(- 5 x + 7) (x - 3)}{{(3 - x)}^{4}}$

Step 4.1.3.5

Cancel the common factor of $x - 3$ and ${(3 - x)}^{4}$ .

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Step 4.1.3.5.1

Factor $- 1$ out of $x$ .

$\frac{(- 5 x + 7) (- 1 (- x) - 3)}{{(3 - x)}^{4}}$

Step 4.1.3.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{(- 5 x + 7) (- 1 (- x) - 1 (3))}{{(3 - x)}^{4}}$

Step 4.1.3.5.3

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

$\frac{(- 5 x + 7) (- 1 (- x + 3))}{{(3 - x)}^{4}}$

Step 4.1.3.5.4

Reorder terms.

$\frac{(- 5 x + 7) (- 1 (- x + 3))}{{(- x + 3)}^{4}}$

Step 4.1.3.5.5

Factor $- x + 3$ out of $(- 5 x + 7) (- 1 (- x + 3))$ .

$\frac{(- x + 3) ((- 5 x + 7) \cdot (- 1))}{{(- x + 3)}^{4}}$

Step 4.1.3.5.6

Cancel the common factors.

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Step 4.1.3.5.6.1

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

$\frac{(- x + 3) ((- 5 x + 7) \cdot (- 1))}{(- x + 3) {(- x + 3)}^{3}}$

Step 4.1.3.5.6.2

Cancel the common factor.

$\frac{(- x + 3) ((- 5 x + 7) \cdot (- 1))}{(- x + 3) {(- x + 3)}^{3}}$

Step 4.1.3.5.6.3

Rewrite the expression.

$\frac{(- 5 x + 7) \cdot (- 1)}{{(- x + 3)}^{3}}$

Step 4.1.3.6

Move $- 1$ to the left of $- 5 x + 7$ .

$\frac{- 1 \cdot (- 5 x + 7)}{{(- x + 3)}^{3}}$

Step 4.1.3.7

Move the negative in front of the fraction.

$- \frac{- 5 x + 7}{{(- x + 3)}^{3}}$

Step 4.1.3.8

Factor $- 1$ out of $- 5 x$ .

$- \frac{- (5 x) + 7}{{(- x + 3)}^{3}}$

Step 4.1.3.9

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

$- \frac{- (5 x) - 1 (- 7)}{{(- x + 3)}^{3}}$

Step 4.1.3.10

Factor $- 1$ out of $- (5 x) - 1 (- 7)$ .

$- \frac{- (5 x - 7)}{{(- x + 3)}^{3}}$

Step 4.1.3.11

Rewrite $- (5 x - 7)$ as $- 1 (5 x - 7)$ .

$- \frac{- 1 (5 x - 7)}{{(- x + 3)}^{3}}$

Step 4.1.3.12

Move the negative in front of the fraction.

$- - \frac{5 x - 7}{{(- x + 3)}^{3}}$

Step 4.1.3.13

Multiply $- 1$ by $- 1$ .

$1 \frac{5 x - 7}{{(- x + 3)}^{3}}$

Step 4.1.3.14

Multiply $\frac{5 x - 7}{{(- x + 3)}^{3}}$ by $1$ .

$f' (x) = \frac{5 x - 7}{{(- x + 3)}^{3}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{5 x - 7}{{(- x + 3)}^{3}}$ .

$\frac{5 x - 7}{{(- x + 3)}^{3}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{5 x - 7}{{(- x + 3)}^{3}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{5 x - 7}{{(- x + 3)}^{3}} = 0$

Step 5.2

Set the numerator equal to zero.

$5 x - 7 = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Add $7$ to both sides of the equation.

$5 x = 7$

Step 5.3.2

Divide each term in $5 x = 7$ by $5$ and simplify.

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Step 5.3.2.1

Divide each term in $5 x = 7$ by $5$ .

$\frac{5 x}{5} = \frac{7}{5}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $5$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{7}{5}$

Step 5.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{5}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{5 x - 7}{{(- x + 3)}^{3}}$ equal to $0$ to find where the expression is undefined.

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

Step 6.2

Solve for $x$ .

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Step 6.2.1

Factor the left side of the equation.

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Step 6.2.1.1

Factor $- 1$ out of $- x + 3$ .

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Step 6.2.1.1.1

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

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

Step 6.2.1.1.2

Rewrite $3$ as $- 1 (- 3)$ .

${(- (x) - 1 \cdot - 3)}^{3} = 0$

Step 6.2.1.1.3

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

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

Step 6.2.1.2

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

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

Step 6.2.2

Divide each term in ${(- 1)}^{3} {(x - 3)}^{3} = 0$ by ${(- 1)}^{3}$ and simplify.

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Step 6.2.2.1

Divide each term in ${(- 1)}^{3} {(x - 3)}^{3} = 0$ by ${(- 1)}^{3}$ .

$\frac{{(- 1)}^{3} {(x - 3)}^{3}}{{(- 1)}^{3}} = \frac{0}{{(- 1)}^{3}}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of ${(- 1)}^{3}$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{{(- 1)}^{3} {(x - 3)}^{3}}{{(- 1)}^{3}} = \frac{0}{{(- 1)}^{3}}$

Step 6.2.2.2.1.2

Divide ${(x - 3)}^{3}$ by $1$ .

${(x - 3)}^{3} = \frac{0}{{(- 1)}^{3}}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

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

${(x - 3)}^{3} = \frac{0}{- 1}$

Step 6.2.2.3.2

Divide $0$ by $- 1$ .

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

Step 6.2.3

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

$x - 3 = 0$

Step 6.2.4

Add $3$ to both sides of the equation.

$x = 3$

Step 7

Critical points to evaluate.

$x = \frac{7}{5}$

Step 8

Evaluate the second derivative at $x = \frac{7}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 (5 (\frac{7}{5}) - 3)}{{(- (\frac{7}{5}) + 3)}^{4}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Cancel the common factor of $5$ .

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Step 9.1.1.1

Cancel the common factor.

$\frac{2 (5 (\frac{7}{5}) - 3)}{{(- \frac{7}{5} + 3)}^{4}}$

Step 9.1.1.2

Rewrite the expression.

$\frac{2 (7 - 3)}{{(- \frac{7}{5} + 3)}^{4}}$

Step 9.1.2

Subtract $3$ from $7$ .

$\frac{2 \cdot 4}{{(- \frac{7}{5} + 3)}^{4}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 \cdot 4}{{(- \frac{7}{5} + 3 \cdot \frac{5}{5})}^{4}}$

Step 9.2.2

Combine $3$ and $\frac{5}{5}$ .

$\frac{2 \cdot 4}{{(- \frac{7}{5} + \frac{3 \cdot 5}{5})}^{4}}$

Step 9.2.3

Combine the numerators over the common denominator.

$\frac{2 \cdot 4}{{(\frac{- 7 + 3 \cdot 5}{5})}^{4}}$

Step 9.2.4

Simplify the numerator.

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Step 9.2.4.1

Multiply $3$ by $5$ .

$\frac{2 \cdot 4}{{(\frac{- 7 + 15}{5})}^{4}}$

Step 9.2.4.2

Add $- 7$ and $15$ .

$\frac{2 \cdot 4}{{(\frac{8}{5})}^{4}}$

Step 9.2.5

Apply the product rule to $\frac{8}{5}$ .

$\frac{2 \cdot 4}{\frac{8^{4}}{5^{4}}}$

Step 9.2.6

Raise $8$ to the power of $4$ .

$\frac{2 \cdot 4}{\frac{4096}{5^{4}}}$

Step 9.2.7

Raise $5$ to the power of $4$ .

$\frac{2 \cdot 4}{\frac{4096}{625}}$

Step 9.3

Multiply $2$ by $4$ .

$\frac{8}{\frac{4096}{625}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$8 (\frac{625}{4096})$

Step 9.5

Cancel the common factor of $8$ .

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Step 9.5.1

Factor $8$ out of $4096$ .

$8 \frac{625}{8 (512)}$

Step 9.5.2

Cancel the common factor.

$8 \frac{625}{8 \cdot 512}$

Step 9.5.3

Rewrite the expression.

$\frac{625}{512}$

Step 10

$x = \frac{7}{5}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{7}{5}$ is a local minimum

Step 11

Find the y-value when $x = \frac{7}{5}$ .

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Step 11.1

Replace the variable $x$ with $\frac{7}{5}$ in the expression.

$f (\frac{7}{5}) = \frac{{(\frac{7}{5})}^{2} - (\frac{7}{5}) - 2}{{(\frac{7}{5})}^{2} - 6 (\frac{7}{5}) + 9}$

Step 11.2

Simplify the result.

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Step 11.2.1

Multiply the numerator and denominator of the fraction by $5$ .

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Step 11.2.1.1

Multiply $\frac{{(\frac{7}{5})}^{2} - \frac{7}{5} - 2}{{(\frac{7}{5})}^{2} - 6 (\frac{7}{5}) + 9}$ by $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{5}{5} \cdot \frac{{(\frac{7}{5})}^{2} - \frac{7}{5} - 2}{{(\frac{7}{5})}^{2} - 6 (\frac{7}{5}) + 9}$

Step 11.2.1.2

Combine.

$f (\frac{7}{5}) = \frac{5 ({(\frac{7}{5})}^{2} - \frac{7}{5} - 2)}{5 ({(\frac{7}{5})}^{2} - 6 (\frac{7}{5}) + 9)}$

Step 11.2.2

Apply the distributive property.

$f (\frac{7}{5}) = \frac{5 {(\frac{7}{5})}^{2} + 5 (- \frac{7}{5}) + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.3

Cancel the common factor of $5$ .

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Step 11.2.3.1

Move the leading negative in $- \frac{7}{5}$ into the numerator.

$f (\frac{7}{5}) = \frac{5 {(\frac{7}{5})}^{2} + 5 (\frac{- 7}{5}) + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.3.2

Cancel the common factor.

$f (\frac{7}{5}) = \frac{5 {(\frac{7}{5})}^{2} + 5 (\frac{- 7}{5}) + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.3.3

Rewrite the expression.

$f (\frac{7}{5}) = \frac{5 {(\frac{7}{5})}^{2} - 7 + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4

Simplify the numerator.

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Step 11.2.4.1

Apply the product rule to $\frac{7}{5}$ .

$f (\frac{7}{5}) = \frac{5 (\frac{7^{2}}{5^{2}}) - 7 + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.2

Cancel the common factor of $5$ .

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Step 11.2.4.2.1

Factor $5$ out of $5^{2}$ .

$f (\frac{7}{5}) = \frac{5 (\frac{7^{2}}{5 \cdot 5}) - 7 + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.2.2

Cancel the common factor.

$f (\frac{7}{5}) = \frac{5 (\frac{7^{2}}{5 \cdot 5}) - 7 + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.2.3

Rewrite the expression.

$f (\frac{7}{5}) = \frac{\frac{7^{2}}{5} - 7 + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.3

Raise $7$ to the power of $2$ .

$f (\frac{7}{5}) = \frac{\frac{49}{5} - 7 + 5 \cdot - 2}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.4

Multiply $5$ by $- 2$ .

$f (\frac{7}{5}) = \frac{\frac{49}{5} - 7 - 10}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.5

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{\frac{49}{5} - 7 \cdot \frac{5}{5} - 10}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.6

Combine $- 7$ and $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{\frac{49}{5} + \frac{- 7 \cdot 5}{5} - 10}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.7

Combine the numerators over the common denominator.

$f (\frac{7}{5}) = \frac{\frac{49 - 7 \cdot 5}{5} - 10}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.8

Simplify the numerator.

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Step 11.2.4.8.1

Multiply $- 7$ by $5$ .

$f (\frac{7}{5}) = \frac{\frac{49 - 35}{5} - 10}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.8.2

Subtract $35$ from $49$ .

$f (\frac{7}{5}) = \frac{\frac{14}{5} - 10}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.9

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{\frac{14}{5} - 10 \cdot \frac{5}{5}}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.10

Combine $- 10$ and $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{\frac{14}{5} + \frac{- 10 \cdot 5}{5}}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.11

Combine the numerators over the common denominator.

$f (\frac{7}{5}) = \frac{\frac{14 - 10 \cdot 5}{5}}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.12

Simplify the numerator.

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Step 11.2.4.12.1

Multiply $- 10$ by $5$ .

$f (\frac{7}{5}) = \frac{\frac{14 - 50}{5}}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.12.2

Subtract $50$ from $14$ .

$f (\frac{7}{5}) = \frac{\frac{- 36}{5}}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.4.13

Move the negative in front of the fraction.

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{5 {(\frac{7}{5})}^{2} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.5

Simplify the denominator.

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Step 11.2.5.1

Apply the product rule to $\frac{7}{5}$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{5 (\frac{7^{2}}{5^{2}}) + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.5.2

Cancel the common factor of $5$ .

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Step 11.2.5.2.1

Factor $5$ out of $5^{2}$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{5 (\frac{7^{2}}{5 \cdot 5}) + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.5.2.2

Cancel the common factor.

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{5 (\frac{7^{2}}{5 \cdot 5}) + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.5.2.3

Rewrite the expression.

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{7^{2}}{5} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.5.3

Raise $7$ to the power of $2$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} + 5 (- 6 (\frac{7}{5})) + 5 \cdot 9}$

Step 11.2.5.4

Multiply $- 6 (\frac{7}{5})$ .

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Step 11.2.5.4.1

Combine $- 6$ and $\frac{7}{5}$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} + 5 (\frac{- 6 \cdot 7}{5}) + 5 \cdot 9}$

Step 11.2.5.4.2

Multiply $- 6$ by $7$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} + 5 (\frac{- 42}{5}) + 5 \cdot 9}$

Step 11.2.5.5

Cancel the common factor of $5$ .

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Step 11.2.5.5.1

Cancel the common factor.

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} + 5 (\frac{- 42}{5}) + 5 \cdot 9}$

Step 11.2.5.5.2

Rewrite the expression.

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} - 42 + 5 \cdot 9}$

Step 11.2.5.6

Multiply $5$ by $9$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} - 42 + 45}$

Step 11.2.5.7

To write $- 42$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} - 42 \cdot \frac{5}{5} + 45}$

Step 11.2.5.8

Combine $- 42$ and $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49}{5} + \frac{- 42 \cdot 5}{5} + 45}$

Step 11.2.5.9

Combine the numerators over the common denominator.

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49 - 42 \cdot 5}{5} + 45}$

Step 11.2.5.10

Simplify the numerator.

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Step 11.2.5.10.1

Multiply $- 42$ by $5$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{49 - 210}{5} + 45}$

Step 11.2.5.10.2

Subtract $210$ from $49$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{- 161}{5} + 45}$

Step 11.2.5.11

To write $45$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{- 161}{5} + 45 \cdot \frac{5}{5}}$

Step 11.2.5.12

Combine $45$ and $\frac{5}{5}$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{- 161}{5} + \frac{45 \cdot 5}{5}}$

Step 11.2.5.13

Combine the numerators over the common denominator.

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{- 161 + 45 \cdot 5}{5}}$

Step 11.2.5.14

Simplify the numerator.

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Step 11.2.5.14.1

Multiply $45$ by $5$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{- 161 + 225}{5}}$

Step 11.2.5.14.2

Add $- 161$ and $225$ .

$f (\frac{7}{5}) = \frac{- \frac{36}{5}}{\frac{64}{5}}$

Step 11.2.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{7}{5}) = - \frac{36}{5} \cdot \frac{5}{64}$

Step 11.2.7

Cancel the common factor of $4$ .

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Step 11.2.7.1

Move the leading negative in $- \frac{36}{5}$ into the numerator.

$f (\frac{7}{5}) = \frac{- 36}{5} \cdot \frac{5}{64}$

Step 11.2.7.2

Factor $4$ out of $- 36$ .

$f (\frac{7}{5}) = \frac{4 (- 9)}{5} \cdot \frac{5}{64}$

Step 11.2.7.3

Factor $4$ out of $64$ .

$f (\frac{7}{5}) = \frac{4 \cdot - 9}{5} \cdot \frac{5}{4 \cdot 16}$

Step 11.2.7.4

Cancel the common factor.

$f (\frac{7}{5}) = \frac{4 \cdot - 9}{5} \cdot \frac{5}{4 \cdot 16}$

Step 11.2.7.5

Rewrite the expression.

$f (\frac{7}{5}) = \frac{- 9}{5} \cdot \frac{5}{16}$

Step 11.2.8

Cancel the common factor of $5$ .

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Step 11.2.8.1

Cancel the common factor.

$f (\frac{7}{5}) = \frac{- 9}{5} \cdot \frac{5}{16}$

Step 11.2.8.2

Rewrite the expression.

$f (\frac{7}{5}) = - 9 (\frac{1}{16})$

Step 11.2.9

Combine $- 9$ and $\frac{1}{16}$ .

$f (\frac{7}{5}) = \frac{- 9}{16}$

Step 11.2.10

Move the negative in front of the fraction.

$f (\frac{7}{5}) = - \frac{9}{16}$

Step 11.2.11

The final answer is $- \frac{9}{16}$ .

$y = - \frac{9}{16}$

Step 12

These are the local extrema for $f (x) = \frac{x^{2} - x - 2}{x^{2} - 6 x + 9}$ .

$(\frac{7}{5}, - \frac{9}{16})$ is a local minima

Step 13