Precalculus Examples

$f (x) = \frac{x + 8}{x - 3}$

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.

$f (x + h) = \frac{(x + h) + 8}{(x + h) - 3}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{x + h + 8}{x + h - 3}$

Step 2.1.2.2

The final answer is $\frac{x + h + 8}{x + h - 3}$ .

$\frac{x + h + 8}{x + h - 3}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{x + h + 8}{x + h - 3}$

$f (x) = \frac{x + 8}{x - 3}$

$f (x + h) = \frac{x + h + 8}{x + h - 3}$

$f (x) = \frac{x + 8}{x - 3}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{x + h + 8}{x + h - 3} - (\frac{x + 8}{x - 3})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{x + h + 8}{x + h - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{\frac{x + h + 8}{x + h - 3} \cdot \frac{x - 3}{x - 3} - \frac{x + 8}{x - 3}}{h}$

Step 4.1.2

To write $- \frac{x + 8}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + h - 3}{x + h - 3}$ .

$\frac{\frac{x + h + 8}{x + h - 3} \cdot \frac{x - 3}{x - 3} - \frac{x + 8}{x - 3} \cdot \frac{x + h - 3}{x + h - 3}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h - 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{x + h + 8}{x + h - 3}$ by $\frac{x - 3}{x - 3}$ .

$\frac{\frac{(x + h + 8) (x - 3)}{(x + h - 3) (x - 3)} - \frac{x + 8}{x - 3} \cdot \frac{x + h - 3}{x + h - 3}}{h}$

Step 4.1.3.2

Multiply $\frac{x + 8}{x - 3}$ by $\frac{x + h - 3}{x + h - 3}$ .

$\frac{\frac{(x + h + 8) (x - 3)}{(x + h - 3) (x - 3)} - \frac{(x + 8) (x + h - 3)}{(x - 3) (x + h - 3)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x - 3) (x + h - 3)$ .

$\frac{\frac{(x + h + 8) (x - 3)}{(x + h - 3) (x - 3)} - \frac{(x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{(x + h + 8) (x - 3) - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5

Rewrite $\frac{(x + h + 8) (x - 3) - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}$ in a factored form.

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Step 4.1.5.1

Expand $(x + h + 8) (x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{x \cdot x + x \cdot - 3 + h x + h \cdot - 3 + 8 x + 8 \cdot - 3 - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.2

Simplify each term.

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Step 4.1.5.2.1

Multiply $x$ by $x$ .

$\frac{\frac{x^{2} + x \cdot - 3 + h x + h \cdot - 3 + 8 x + 8 \cdot - 3 - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.2.2

Move $- 3$ to the left of $x$ .

$\frac{\frac{x^{2} - 3 \cdot x + h x + h \cdot - 3 + 8 x + 8 \cdot - 3 - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.2.3

Move $- 3$ to the left of $h$ .

$\frac{\frac{x^{2} - 3 x + h x - 3 \cdot h + 8 x + 8 \cdot - 3 - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.2.4

Multiply $8$ by $- 3$ .

$\frac{\frac{x^{2} - 3 x + h x - 3 h + 8 x - 24 - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.3

Add $- 3 x$ and $8 x$ .

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 - (x + 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.4

Apply the distributive property.

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 + (- x - 1 \cdot 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.5

Multiply $- 1$ by $8$ .

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 + (- x - 8) (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.6

Expand $(- x - 8) (x + h - 3)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 - x \cdot x - x h - x \cdot - 3 - 8 x - 8 h - 8 \cdot - 3}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.7

Simplify each term.

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Step 4.1.5.7.1

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

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Step 4.1.5.7.1.1

Move $x$ .

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 - (x \cdot x) - x h - x \cdot - 3 - 8 x - 8 h - 8 \cdot - 3}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.7.1.2

Multiply $x$ by $x$ .

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 - x^{2} - x h - x \cdot - 3 - 8 x - 8 h - 8 \cdot - 3}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.7.2

Multiply $- 3$ by $- 1$ .

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 - x^{2} - x h + 3 x - 8 x - 8 h - 8 \cdot - 3}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.7.3

Multiply $- 8$ by $- 3$ .

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 - x^{2} - x h + 3 x - 8 x - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.8

Subtract $8 x$ from $3 x$ .

$\frac{\frac{x^{2} + 5 x + h x - 3 h - 24 - x^{2} - x h - 5 x - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.9

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

$\frac{\frac{5 x + h x - 3 h - 24 + 0 - x h - 5 x - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.10

Add $5 x$ and $0$ .

$\frac{\frac{5 x + h x - 3 h - 24 - x h - 5 x - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.11

Subtract $5 x$ from $5 x$ .

$\frac{\frac{h x - 3 h - 24 - x h + 0 - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.12

Subtract $x h$ from $h x$ .

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Step 4.1.5.12.1

Reorder $h$ and $x$ .

$\frac{\frac{- 3 h - 24 + x h - x h + 0 - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.12.2

Subtract $x h$ from $x h$ .

$\frac{\frac{- 3 h - 24 + 0 + 0 - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.13

Add $- 3 h$ and $0$ .

$\frac{\frac{- 3 h - 24 + 0 - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.14

Add $- 3 h$ and $0$ .

$\frac{\frac{- 3 h - 24 - 8 h + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.15

Subtract $8 h$ from $- 3 h$ .

$\frac{\frac{- 11 h - 24 + 24}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.16

Add $- 24$ and $24$ .

$\frac{\frac{- 11 h + 0}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.17

Add $- 11 h$ and $0$ .

$\frac{\frac{- 11 h}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{11 h}{(x + h - 3) (x - 3)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{11 h}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{11 h}{(x + h - 3) (x - 3)}$ into the numerator.

$\frac{- 11 h}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 11 h$ .

$\frac{h \cdot - 11}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 11}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 11}{(x + h - 3) (x - 3)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{11}{(x + h - 3) (x - 3)}$

Step 5