Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

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

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

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

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Step 4.1.5.1

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

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

Step 4.1.5.2

Combine the opposite terms in $x \cdot x + x \cdot 3 + h x + h \cdot 3 - 3 x - 3 \cdot 3$ .

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Step 4.1.5.2.1

Reorder the factors in the terms $x \cdot 3$ and $- 3 x$ .

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

Step 4.1.5.2.2

Subtract $3 x$ from $3 x$ .

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

Step 4.1.5.2.3

Add $x \cdot x + h x + h \cdot 3$ and $0$ .

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

Step 4.1.5.3

Simplify each term.

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Step 4.1.5.3.1

Multiply $x$ by $x$ .

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

Step 4.1.5.3.2

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

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

Step 4.1.5.3.3

Multiply $- 3$ by $3$ .

$\frac{\frac{x^{2} + h x + 3 h - 9 - (x - 3) (x + h + 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.4

Apply the distributive property.

$\frac{\frac{x^{2} + h x + 3 h - 9 + (- x - - 3) (x + h + 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.5

Multiply $- 1$ by $- 3$ .

$\frac{\frac{x^{2} + h x + 3 h - 9 + (- x + 3) (x + h + 3)}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.6

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

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

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 $x$ .

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

Step 4.1.5.7.1.2

Multiply $x$ by $x$ .

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

Step 4.1.5.7.2

Multiply $3$ by $- 1$ .

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

Step 4.1.5.7.3

Multiply $3$ by $3$ .

$\frac{\frac{x^{2} + h x + 3 h - 9 - x^{2} - x h - 3 x + 3 x + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.8

Combine the opposite terms in $- x^{2} - x h - 3 x + 3 x + 3 h + 9$ .

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Step 4.1.5.8.1

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

$\frac{\frac{x^{2} + h x + 3 h - 9 - x^{2} - x h + 0 + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.8.2

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

$\frac{\frac{x^{2} + h x + 3 h - 9 - x^{2} - x h + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.9

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

$\frac{\frac{h x + 3 h - 9 + 0 - x h + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.10

Add $h x$ and $0$ .

$\frac{\frac{h x + 3 h - 9 - x h + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.11

Subtract $x h$ from $h x$ .

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Step 4.1.5.11.1

Reorder $h$ and $x$ .

$\frac{\frac{3 h - 9 + x h - x h + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.11.2

Subtract $x h$ from $x h$ .

$\frac{\frac{3 h - 9 + 0 + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.12

Add $3 h$ and $0$ .

$\frac{\frac{3 h - 9 + 3 h + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.13

Add $3 h$ and $3 h$ .

$\frac{\frac{6 h - 9 + 9}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.14

Add $- 9$ and $9$ .

$\frac{\frac{6 h + 0}{(x + 3) (x + h + 3)}}{h}$

Step 4.1.5.15

Add $6 h$ and $0$ .

$\frac{\frac{6 h}{(x + 3) (x + h + 3)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 h}{(x + 3) (x + h + 3)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $6 h$ .

$\frac{h \cdot 6}{(x + 3) (x + h + 3)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$\frac{h \cdot 6}{(x + 3) (x + h + 3)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$\frac{6}{(x + 3) (x + h + 3)}$

Step 5