Precalculus Examples

$f (x) = - \frac{3}{x - 4}$

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.

$f (x + h) = - \frac{3}{(x + h) - 4}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = - \frac{3}{x + h - 4}$

Step 2.1.2.2

The final answer is $- \frac{3}{x + h - 4}$ .

$- \frac{3}{x + h - 4}$

Step 2.2

Find the components of the definition.

$f (x + h) = - \frac{3}{x + h - 4}$

$f (x) = - \frac{3}{x - 4}$

$f (x + h) = - \frac{3}{x + h - 4}$

$f (x) = - \frac{3}{x - 4}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{3}{x + h - 4} - (- \frac{3}{x - 4})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- - \frac{3}{x - 4}$ .

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Step 4.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{3}{x + h - 4} + 1 \frac{3}{x - 4}}{h}$

Step 4.1.1.2

Multiply $\frac{3}{x - 4}$ by $1$ .

$\frac{- \frac{3}{x + h - 4} + \frac{3}{x - 4}}{h}$

Step 4.1.2

To write $- \frac{3}{x + h - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{- \frac{3}{x + h - 4} \cdot \frac{x - 4}{x - 4} + \frac{3}{x - 4}}{h}$

Step 4.1.3

To write $\frac{3}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + h - 4}{x + h - 4}$ .

$\frac{- \frac{3}{x + h - 4} \cdot \frac{x - 4}{x - 4} + \frac{3}{x - 4} \cdot \frac{x + h - 4}{x + h - 4}}{h}$

Step 4.1.4

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 4.1.4.1

Multiply $\frac{3}{x + h - 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{- \frac{3 (x - 4)}{(x + h - 4) (x - 4)} + \frac{3}{x - 4} \cdot \frac{x + h - 4}{x + h - 4}}{h}$

Step 4.1.4.2

Multiply $\frac{3}{x - 4}$ by $\frac{x + h - 4}{x + h - 4}$ .

$\frac{- \frac{3 (x - 4)}{(x + h - 4) (x - 4)} + \frac{3 (x + h - 4)}{(x - 4) (x + h - 4)}}{h}$

Step 4.1.4.3

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

$\frac{- \frac{3 (x - 4)}{(x + h - 4) (x - 4)} + \frac{3 (x + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{\frac{- 3 (x - 4) + 3 (x + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6

Rewrite $\frac{- 3 (x - 4) + 3 (x + h - 4)}{(x + h - 4) (x - 4)}$ in a factored form.

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Step 4.1.6.1

Factor $3$ out of $- 3 (x - 4) + 3 (x + h - 4)$ .

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Step 4.1.6.1.1

Factor $3$ out of $- 3 (x - 4)$ .

$\frac{\frac{3 (- (x - 4)) + 3 (x + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6.1.2

Factor $3$ out of $3 (- (x - 4)) + 3 (x + h - 4)$ .

$\frac{\frac{3 (- (x - 4) + x + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6.2

Apply the distributive property.

$\frac{\frac{3 (- x - - 4 + x + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6.3

Multiply $- 1$ by $- 4$ .

$\frac{\frac{3 (- x + 4 + x + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6.4

Add $- x$ and $x$ .

$\frac{\frac{3 (0 + 4 + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6.5

Add $0$ and $4$ .

$\frac{\frac{3 (4 + h - 4)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6.6

Subtract $4$ from $4$ .

$\frac{\frac{3 (h + 0)}{(x + h - 4) (x - 4)}}{h}$

Step 4.1.6.7

Add $h$ and $0$ .

$\frac{\frac{3 h}{(x + h - 4) (x - 4)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 h}{(x + h - 4) (x - 4)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $3 h$ .

$\frac{h \cdot 3}{(x + h - 4) (x - 4)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$\frac{h \cdot 3}{(x + h - 4) (x - 4)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$\frac{3}{(x + h - 4) (x - 4)}$

Step 5