Precalculus Examples

$f (x) = \frac{4}{x + 4}$

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.

$f (x + h) = \frac{4}{(x + h) + 4}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{4}{x + h + 4}$

Step 2.1.2.2

The final answer is $\frac{4}{x + h + 4}$ .

$\frac{4}{x + h + 4}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{4}{x + h + 4}$

$f (x) = \frac{4}{x + 4}$

$f (x + h) = \frac{4}{x + h + 4}$

$f (x) = \frac{4}{x + 4}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{4}{x + h + 4} - (\frac{4}{x + 4})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{4}{x + h + 4}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{\frac{4}{x + h + 4} \cdot \frac{x + 4}{x + 4} - \frac{4}{x + 4}}{h}$

Step 4.1.2

To write $- \frac{4}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x + h + 4}{x + h + 4}$ .

$\frac{\frac{4}{x + h + 4} \cdot \frac{x + 4}{x + 4} - \frac{4}{x + 4} \cdot \frac{x + h + 4}{x + h + 4}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h + 4) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{4}{x + h + 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{\frac{4 (x + 4)}{(x + h + 4) (x + 4)} - \frac{4}{x + 4} \cdot \frac{x + h + 4}{x + h + 4}}{h}$

Step 4.1.3.2

Multiply $\frac{4}{x + 4}$ by $\frac{x + h + 4}{x + h + 4}$ .

$\frac{\frac{4 (x + 4)}{(x + h + 4) (x + 4)} - \frac{4 (x + h + 4)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x + h + 4) (x + 4)$ .

$\frac{\frac{4 (x + 4)}{(x + 4) (x + h + 4)} - \frac{4 (x + h + 4)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{4 (x + 4) - 4 (x + h + 4)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5

Rewrite $\frac{4 (x + 4) - 4 (x + h + 4)}{(x + 4) (x + h + 4)}$ in a factored form.

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Step 4.1.5.1

Factor $4$ out of $4 (x + 4) - 4 (x + h + 4)$ .

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Step 4.1.5.1.1

Factor $4$ out of $- 4 (x + h + 4)$ .

$\frac{\frac{4 (x + 4) + 4 (- (x + h + 4))}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.1.2

Factor $4$ out of $4 (x + 4) + 4 (- (x + h + 4))$ .

$\frac{\frac{4 (x + 4 - (x + h + 4))}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.2

Apply the distributive property.

$\frac{\frac{4 (x + 4 - x - h - 1 \cdot 4)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.3

Multiply $- 1$ by $4$ .

$\frac{\frac{4 (x + 4 - x - h - 4)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.4

Subtract $x$ from $x$ .

$\frac{\frac{4 (0 + 4 - h - 4)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.5

Add $0$ and $4$ .

$\frac{\frac{4 (4 - h - 4)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.6

Subtract $4$ from $4$ .

$\frac{\frac{4 (- h + 0)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.7

Add $- h$ and $0$ .

$\frac{\frac{4 \cdot - 1 h}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.8

Combine exponents.

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Step 4.1.5.8.1

Factor out negative.

$\frac{\frac{- (4 h)}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.5.8.2

Multiply $4$ by $- 1$ .

$\frac{\frac{- 4 h}{(x + 4) (x + h + 4)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{4 h}{(x + 4) (x + h + 4)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{4 h}{(x + 4) (x + h + 4)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{4 h}{(x + 4) (x + h + 4)}$ into the numerator.

$\frac{- 4 h}{(x + 4) (x + h + 4)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 4 h$ .

$\frac{h \cdot - 4}{(x + 4) (x + h + 4)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 4}{(x + 4) (x + h + 4)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 4}{(x + 4) (x + h + 4)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{4}{(x + 4) (x + h + 4)}$

Step 5