Algebra Examples

$f (x) = \frac{x^{2} - 81}{x^{2} - 11 x + 18}$

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 $f (x) = x^{2} - 81$ and $g (x) = x^{2} - 11 x + 18$ .

$\frac{(x^{2} - 11 x + 18) \frac{d}{d x} [x^{2} - 81] - (x^{2} - 81) \frac{d}{d x} [x^{2} - 11 x + 18]}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} - 81$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 81]$ .

$\frac{(x^{2} - 11 x + 18) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 81]) - (x^{2} - 81) \frac{d}{d x} [x^{2} - 11 x + 18]}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.2

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

$\frac{(x^{2} - 11 x + 18) (2 x + \frac{d}{d x} [- 81]) - (x^{2} - 81) \frac{d}{d x} [x^{2} - 11 x + 18]}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.3

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81$ with respect to $x$ is $0$ .

$\frac{(x^{2} - 11 x + 18) (2 x + 0) - (x^{2} - 81) \frac{d}{d x} [x^{2} - 11 x + 18]}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $2 x$ and $0$ .

$\frac{(x^{2} - 11 x + 18) (2 x) - (x^{2} - 81) \frac{d}{d x} [x^{2} - 11 x + 18]}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.4.2

Move $2$ to the left of $x^{2} - 11 x + 18$ .

$\frac{2 \cdot (x^{2} - 11 x + 18) x - (x^{2} - 81) \frac{d}{d x} [x^{2} - 11 x + 18]}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.5

By the Sum Rule, the derivative of $x^{2} - 11 x + 18$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [18]$ .

$\frac{2 (x^{2} - 11 x + 18) x - (x^{2} - 81) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [18])}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.6

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

$\frac{2 (x^{2} - 11 x + 18) x - (x^{2} - 81) (2 x + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [18])}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.7

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$\frac{2 (x^{2} - 11 x + 18) x - (x^{2} - 81) (2 x - 11 \frac{d}{d x} [x] + \frac{d}{d x} [18])}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.8

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

$\frac{2 (x^{2} - 11 x + 18) x - (x^{2} - 81) (2 x - 11 \cdot 1 + \frac{d}{d x} [18])}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.9

Multiply $- 11$ by $1$ .

$\frac{2 (x^{2} - 11 x + 18) x - (x^{2} - 81) (2 x - 11 + \frac{d}{d x} [18])}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.10

Since $18$ is constant with respect to $x$ , the derivative of $18$ with respect to $x$ is $0$ .

$\frac{2 (x^{2} - 11 x + 18) x - (x^{2} - 81) (2 x - 11 + 0)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.2.11

Add $2 x - 11$ and $0$ .

$\frac{2 (x^{2} - 11 x + 18) x - (x^{2} - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 (- 11 x) + 2 \cdot 18) x - (x^{2} - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 (- 11 x) x + 2 \cdot 18 x - (x^{2} - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.3

Apply the distributive property.

$\frac{2 x^{2} x + 2 (- 11 x) x + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

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

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Step 1.3.4.1.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 (- 11 x) x + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.1.2

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

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Step 1.3.4.1.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 2 (- 11 x) x + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.1.2.2

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

$\frac{2 x^{1 + 2} + 2 (- 11 x) x + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 (- 11 x) x + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.2

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

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Step 1.3.4.1.2.1

Move $x$ .

$\frac{2 x^{3} + 2 (- 11 (x \cdot x)) + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 2 (- 11 x^{2}) + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.3

Multiply $- 11$ by $2$ .

$\frac{2 x^{3} - 22 x^{2} + 2 \cdot 18 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.4

Multiply $2$ by $18$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x + (- x^{2} - - 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.5

Multiply $- 1$ by $- 81$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x + (- x^{2} + 81) (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.6

Expand $(- x^{2} + 81) (2 x - 11)$ using the FOIL Method.

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Step 1.3.4.1.6.1

Apply the distributive property.

$\frac{2 x^{3} - 22 x^{2} + 36 x - x^{2} (2 x - 11) + 81 (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.6.2

Apply the distributive property.

$\frac{2 x^{3} - 22 x^{2} + 36 x - x^{2} (2 x) - x^{2} \cdot - 11 + 81 (2 x - 11)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.6.3

Apply the distributive property.

$\frac{2 x^{3} - 22 x^{2} + 36 x - x^{2} (2 x) - x^{2} \cdot - 11 + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7

Simplify each term.

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Step 1.3.4.1.7.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 22 x^{2} + 36 x - 1 \cdot 2 x^{2} x - x^{2} \cdot - 11 + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.2

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

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Step 1.3.4.1.7.2.1

Move $x$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot - 11 + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.2.2

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

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Step 1.3.4.1.7.2.2.1

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

$\frac{2 x^{3} - 22 x^{2} + 36 x - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot - 11 + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.2.2.2

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

$\frac{2 x^{3} - 22 x^{2} + 36 x - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot - 11 + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x - 1 \cdot 2 x^{3} - x^{2} \cdot - 11 + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x - 2 x^{3} - x^{2} \cdot - 11 + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.4

Multiply $- 11$ by $- 1$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x - 2 x^{3} + 11 x^{2} + 81 (2 x) + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.5

Multiply $2$ by $81$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x - 2 x^{3} + 11 x^{2} + 162 x + 81 \cdot - 11}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.1.7.6

Multiply $81$ by $- 11$ .

$\frac{2 x^{3} - 22 x^{2} + 36 x - 2 x^{3} + 11 x^{2} + 162 x - 891}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.2

Combine the opposite terms in $2 x^{3} - 22 x^{2} + 36 x - 2 x^{3} + 11 x^{2} + 162 x - 891$ .

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Step 1.3.4.2.1

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

$\frac{- 22 x^{2} + 36 x + 0 + 11 x^{2} + 162 x - 891}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.2.2

Add $- 22 x^{2} + 36 x$ and $0$ .

$\frac{- 22 x^{2} + 36 x + 11 x^{2} + 162 x - 891}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.3

Add $- 22 x^{2}$ and $11 x^{2}$ .

$\frac{- 11 x^{2} + 36 x + 162 x - 891}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.4.4

Add $36 x$ and $162 x$ .

$\frac{- 11 x^{2} + 198 x - 891}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5

Simplify the numerator.

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Step 1.3.5.1

Factor $11$ out of $- 11 x^{2} + 198 x - 891$ .

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Step 1.3.5.1.1

Factor $11$ out of $- 11 x^{2}$ .

$\frac{11 (- x^{2}) + 198 x - 891}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.1.2

Factor $11$ out of $198 x$ .

$\frac{11 (- x^{2}) + 11 (18 x) - 891}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.1.3

Factor $11$ out of $- 891$ .

$\frac{11 (- x^{2}) + 11 (18 x) + 11 (- 81)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.1.4

Factor $11$ out of $11 (- x^{2}) + 11 (18 x)$ .

$\frac{11 (- x^{2} + 18 x) + 11 (- 81)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.1.5

Factor $11$ out of $11 (- x^{2} + 18 x) + 11 (- 81)$ .

$\frac{11 (- x^{2} + 18 x - 81)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.2

Factor by grouping.

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Step 1.3.5.2.1

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 $a \cdot c = - 1 \cdot - 81 = 81$ and whose sum is $b = 18$ .

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Step 1.3.5.2.1.1

Factor $18$ out of $18 x$ .

$\frac{11 (- x^{2} + 18 (x) - 81)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.2.1.2

Rewrite $18$ as $9$ plus $9$

$\frac{11 (- x^{2} + (9 + 9) x - 81)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.2.1.3

Apply the distributive property.

$\frac{11 (- x^{2} + 9 x + 9 x - 81)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.2.2

Factor out the greatest common factor from each group.

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Step 1.3.5.2.2.1

Group the first two terms and the last two terms.

$\frac{11 ((- x^{2} + 9 x) + 9 x - 81)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.2.2.2

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

$\frac{11 (x (- x + 9) - 9 (- x + 9))}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 9$ .

$\frac{11 ((- x + 9) (x - 9))}{{(x^{2} - 11 x + 18)}^{2}}$

$\frac{11 (- x + 9) (x - 9)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3

Combine exponents.

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Step 1.3.5.3.1

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

$\frac{11 (- (x) + 9) (x - 9)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.2

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{11 (- (x) - 1 (- 9)) (x - 9)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.3

Factor $- 1$ out of $- (x) - 1 (- 9)$ .

$\frac{11 (- (x - 9)) (x - 9)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.4

Rewrite $- (x - 9)$ as $- 1 (x - 9)$ .

$\frac{11 (- 1 (x - 9)) (x - 9)}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.5

Raise $x - 9$ to the power of $1$ .

$\frac{11 \cdot - 1 ({(x - 9)}^{1} (x - 9))}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.6

Raise $x - 9$ to the power of $1$ .

$\frac{11 \cdot - 1 ({(x - 9)}^{1} {(x - 9)}^{1})}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.7

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

$\frac{11 \cdot - 1 {(x - 9)}^{1 + 1}}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.8

Add $1$ and $1$ .

$\frac{11 \cdot - 1 {(x - 9)}^{2}}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.5.3.9

Multiply $11$ by $- 1$ .

$\frac{- 11 {(x - 9)}^{2}}{{(x^{2} - 11 x + 18)}^{2}}$

Step 1.3.6

Simplify the denominator.

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Step 1.3.6.1

Factor $x^{2} - 11 x + 18$ using the AC method.

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Step 1.3.6.1.1

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 $18$ and whose sum is $- 11$ .

$- 9, - 2$

Step 1.3.6.1.2

Write the factored form using these integers.

$\frac{- 11 {(x - 9)}^{2}}{{((x - 9) (x - 2))}^{2}}$

Step 1.3.6.2

Apply the product rule to $(x - 9) (x - 2)$ .

$\frac{- 11 {(x - 9)}^{2}}{{(x - 9)}^{2} {(x - 2)}^{2}}$

Step 1.3.7

Cancel the common factor of ${(x - 9)}^{2}$ .

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Step 1.3.7.1

Cancel the common factor.

$\frac{- 11 {(x - 9)}^{2}}{{(x - 9)}^{2} {(x - 2)}^{2}}$

Step 1.3.7.2

Rewrite the expression.

$\frac{- 11}{{(x - 2)}^{2}}$

Step 1.3.8

Move the negative in front of the fraction.

$- \frac{11}{{(x - 2)}^{2}}$

Step 2

Find the second 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 $- 11$ is constant with respect to $x$ , the derivative of $- \frac{11}{{(x - 2)}^{2}}$ with respect to $x$ is $- 11 \frac{d}{d x} [\frac{1}{{(x - 2)}^{2}}]$ .

$f'' (x) = - 11 \frac{d}{d x} \frac{1}{{(x - 2)}^{2}}$

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

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

$f'' (x) = - 11 \frac{d}{d x} {({(x - 2)}^{2})}^{- 1}$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$f'' (x) = - 11 \frac{d}{d x} {(x - 2)}^{2 \cdot - 1}$

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

$f'' (x) = - 11 \frac{d}{d x} {(x - 2)}^{- 2}$

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x - 2$ .

$f'' (x) = - 11 (\frac{d}{d u} (u^{- 2}) \frac{d}{d x} (x - 2))$

Step 2.2.2

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

$f'' (x) = - 11 (- 2 u^{- 3} \frac{d}{d x} (x - 2))$

Step 2.2.3

Replace all occurrences of $u$ with $x - 2$ .

$f'' (x) = - 11 (- 2 {(x - 2)}^{- 3} \frac{d}{d x} (x - 2))$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 2$ by $- 11$ .

$f'' (x) = 22 ({(x - 2)}^{- 3} \frac{d}{d x} (x - 2))$

Step 2.3.2

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

$f'' (x) = 22 {(x - 2)}^{- 3} (\frac{d}{d x} (x) + \frac{d}{d x} (- 2))$

Step 2.3.3

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

$f'' (x) = 22 {(x - 2)}^{- 3} (1 + \frac{d}{d x} (- 2))$

Step 2.3.4

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

$f'' (x) = 22 {(x - 2)}^{- 3} (1 + 0)$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add $1$ and $0$ .

$f'' (x) = 22 {(x - 2)}^{- 3} \cdot 1$

Step 2.3.5.2

Multiply $22$ by $1$ .

$f'' (x) = 22 {(x - 2)}^{- 3}$

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = 22 (\frac{1}{{(x - 2)}^{3}})$

Step 2.4.2

Combine $22$ and $\frac{1}{{(x - 2)}^{3}}$ .

$f'' (x) = \frac{22}{{(x - 2)}^{3}}$

Step 3

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

$- \frac{11}{{(x - 2)}^{2}} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6