Precalculus Examples

$f (x) = \frac{- 1}{x^{2} - 36}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \frac{- 1}{{(x + h)}^{2} - 36}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

Rewrite $36$ as $6^{2}$ .

$f (x + h) = \frac{- 1}{{(x + h)}^{2} - 6^{2}}$

Step 2.1.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + h$ and $b = 6$ .

$f (x + h) = \frac{- 1}{(x + h + 6) (x + h - 6)}$

Step 2.1.2.2

Move the negative in front of the fraction.

$f (x + h) = - \frac{1}{(x + h + 6) (x + h - 6)}$

Step 2.1.2.3

The final answer is $- \frac{1}{(x + h + 6) (x + h - 6)}$ .

$- \frac{1}{(x + h + 6) (x + h - 6)}$

Step 2.2

Find the components of the definition.

$f (x + h) = - \frac{1}{(x + h + 6) (x + h - 6)}$

$f (x) = - \frac{1}{(x + 6) (x - 6)}$

$f (x + h) = - \frac{1}{(x + h + 6) (x + h - 6)}$

$f (x) = - \frac{1}{(x + 6) (x - 6)}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{1}{(x + h + 6) (x + h - 6)} - (- \frac{1}{(x + 6) (x - 6)})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- - \frac{1}{(x + 6) (x - 6)}$ .

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Step 4.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{1}{(x + h + 6) (x + h - 6)} + 1 \frac{1}{(x + 6) (x - 6)}}{h}$

Step 4.1.1.2

Multiply $\frac{1}{(x + 6) (x - 6)}$ by $1$ .

$\frac{- \frac{1}{(x + h + 6) (x + h - 6)} + \frac{1}{(x + 6) (x - 6)}}{h}$

Step 4.1.2

To write $- \frac{1}{(x + h + 6) (x + h - 6)}$ as a fraction with a common denominator, multiply by $\frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$ .

$\frac{- \frac{1}{(x + h + 6) (x + h - 6)} \cdot \frac{(x + 6) (x - 6)}{(x + 6) (x - 6)} + \frac{1}{(x + 6) (x - 6)}}{h}$

Step 4.1.3

To write $\frac{1}{(x + 6) (x - 6)}$ as a fraction with a common denominator, multiply by $\frac{(x + h + 6) (x + h - 6)}{(x + h + 6) (x + h - 6)}$ .

$\frac{- \frac{1}{(x + h + 6) (x + h - 6)} \cdot \frac{(x + 6) (x - 6)}{(x + 6) (x - 6)} + \frac{1}{(x + 6) (x - 6)} \cdot \frac{(x + h + 6) (x + h - 6)}{(x + h + 6) (x + h - 6)}}{h}$

Step 4.1.4

Write each expression with a common denominator of $(x + h + 6) (x + h - 6) (x + 6) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.4.1

Multiply $\frac{1}{(x + h + 6) (x + h - 6)}$ by $\frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$ .

$\frac{- \frac{(x + 6) (x - 6)}{(x + h + 6) (x + h - 6) ((x + 6) (x - 6))} + \frac{1}{(x + 6) (x - 6)} \cdot \frac{(x + h + 6) (x + h - 6)}{(x + h + 6) (x + h - 6)}}{h}$

Step 4.1.4.2

Multiply $\frac{1}{(x + 6) (x - 6)}$ by $\frac{(x + h + 6) (x + h - 6)}{(x + h + 6) (x + h - 6)}$ .

$\frac{- \frac{(x + 6) (x - 6)}{(x + h + 6) (x + h - 6) ((x + 6) (x - 6))} + \frac{(x + h + 6) (x + h - 6)}{(x + 6) (x - 6) ((x + h + 6) (x + h - 6))}}{h}$

Step 4.1.4.3

Reorder the factors of $(x + h + 6) (x + h - 6) ((x + 6) (x - 6))$ .

$\frac{- \frac{(x + 6) (x - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)} + \frac{(x + h + 6) (x + h - 6)}{(x + 6) (x - 6) ((x + h + 6) (x + h - 6))}}{h}$

Step 4.1.4.4

Reorder the factors of $(x + 6) (x - 6) ((x + h + 6) (x + h - 6))$ .

$\frac{- \frac{(x + 6) (x - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)} + \frac{(x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{\frac{- (x + 6) (x - 6) + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6

Simplify the numerator.

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Step 4.1.6.1

Apply the distributive property.

$\frac{\frac{(- x - 1 \cdot 6) (x - 6) + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.2

Multiply $- 1$ by $6$ .

$\frac{\frac{(- x - 6) (x - 6) + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.3

Expand $(- x - 6) (x - 6)$ using the FOIL Method.

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Step 4.1.6.3.1

Apply the distributive property.

$\frac{\frac{- x (x - 6) - 6 (x - 6) + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.3.2

Apply the distributive property.

$\frac{\frac{- x \cdot x - x \cdot - 6 - 6 (x - 6) + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.3.3

Apply the distributive property.

$\frac{\frac{- x \cdot x - x \cdot - 6 - 6 x - 6 \cdot - 6 + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.4

Simplify and combine like terms.

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Step 4.1.6.4.1

Simplify each term.

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Step 4.1.6.4.1.1

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

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Step 4.1.6.4.1.1.1

Move $x$ .

$\frac{\frac{- (x \cdot x) - x \cdot - 6 - 6 x - 6 \cdot - 6 + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.4.1.1.2

Multiply $x$ by $x$ .

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

Step 4.1.6.4.1.2

Multiply $- 6$ by $- 1$ .

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

Step 4.1.6.4.1.3

Multiply $- 6$ by $- 6$ .

$\frac{\frac{- x^{2} + 6 x - 6 x + 36 + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.4.2

Subtract $6 x$ from $6 x$ .

$\frac{\frac{- x^{2} + 0 + 36 + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.4.3

Add $- x^{2}$ and $0$ .

$\frac{\frac{- x^{2} + 36 + (x + h + 6) (x + h - 6)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.5

Expand $(x + h + 6) (x + h - 6)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{- x^{2} + 36 + x \cdot x + x h + x \cdot - 6 + h x + h \cdot h + h \cdot - 6 + 6 x + 6 h + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.6

Combine the opposite terms in $x \cdot x + x h + x \cdot - 6 + h x + h \cdot h + h \cdot - 6 + 6 x + 6 h + 6 \cdot - 6$ .

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Step 4.1.6.6.1

Reorder the factors in the terms $x \cdot - 6$ and $6 x$ .

$\frac{\frac{- x^{2} + 36 + x \cdot x + x h - 6 x + h x + h \cdot h + h \cdot - 6 + 6 x + 6 h + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.6.2

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

$\frac{\frac{- x^{2} + 36 + x \cdot x + x h + h x + h \cdot h + h \cdot - 6 + 0 + 6 h + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.6.3

Add $x \cdot x + x h + h x + h \cdot h + h \cdot - 6$ and $0$ .

$\frac{\frac{- x^{2} + 36 + x \cdot x + x h + h x + h \cdot h + h \cdot - 6 + 6 h + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.6.4

Reorder the factors in the terms $h \cdot - 6$ and $6 h$ .

$\frac{\frac{- x^{2} + 36 + x \cdot x + x h + h x + h \cdot h - 6 h + 6 h + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.6.5

Add $- 6 h$ and $6 h$ .

$\frac{\frac{- x^{2} + 36 + x \cdot x + x h + h x + h \cdot h + 0 + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.6.6

Add $x \cdot x + x h + h x + h \cdot h$ and $0$ .

$\frac{\frac{- x^{2} + 36 + x \cdot x + x h + h x + h \cdot h + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.7

Simplify each term.

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Step 4.1.6.7.1

Multiply $x$ by $x$ .

$\frac{\frac{- x^{2} + 36 + x^{2} + x h + h x + h \cdot h + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.7.2

Multiply $h$ by $h$ .

$\frac{\frac{- x^{2} + 36 + x^{2} + x h + h x + h^{2} + 6 \cdot - 6}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.7.3

Multiply $6$ by $- 6$ .

$\frac{\frac{- x^{2} + 36 + x^{2} + x h + h x + h^{2} - 36}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.8

Add $x h$ and $h x$ .

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Step 4.1.6.8.1

Reorder $h$ and $x$ .

$\frac{\frac{- x^{2} + 36 + x^{2} + x h + x h + h^{2} - 36}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.8.2

Add $x h$ and $x h$ .

$\frac{\frac{- x^{2} + 36 + x^{2} + 2 x h + h^{2} - 36}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.9

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

$\frac{\frac{0 + 36 + 2 x h + h^{2} - 36}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.10

Add $0$ and $36$ .

$\frac{\frac{36 + 2 x h + h^{2} - 36}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.11

Subtract $36$ from $36$ .

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

Step 4.1.6.12

Add $2 x h + h^{2}$ and $0$ .

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

Step 4.1.6.13

Factor $h$ out of $2 x h + h^{2}$ .

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Step 4.1.6.13.1

Factor $h$ out of $2 x h$ .

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

Step 4.1.6.13.2

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

$\frac{\frac{h (2 x) + h (h)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.1.6.13.3

Factor $h$ out of $h (2 x) + h (h)$ .

$\frac{\frac{h (2 x + h)}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{2 x + h}{(x + 6) (x + h + 6) (x + h - 6) (x - 6)}$

Step 5