Find the Difference Quotient f(x)=1/(x+8)

$f (x) = \frac{1}{x + 8}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

$f (x + h) = \frac{1}{(x + h) + 8}$

Simplify the result.

Remove parentheses.

$f (x + h) = \frac{1}{x + h + 8}$

The final answer is $\frac{1}{x + h + 8}$ .

$\frac{1}{x + h + 8}$

Find the components of the definition.

$f (x + h) = \frac{1}{x + h + 8}$

$f (x) = \frac{1}{x + 8}$

$f (x + h) = \frac{1}{x + h + 8}$

$f (x) = \frac{1}{x + 8}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{1}{x + h + 8} - (\frac{1}{x + 8})}{h}$

Step 4

Simplify.

Simplify the numerator.

To write $\frac{1}{x + h + 8}$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

$\frac{\frac{1}{x + h + 8} \cdot \frac{x + 8}{x + 8} - \frac{1}{x + 8}}{h}$

To write $- \frac{1}{x + 8}$ as a fraction with a common denominator, multiply by $\frac{x + h + 8}{x + h + 8}$ .

$\frac{\frac{1}{x + h + 8} \cdot \frac{x + 8}{x + 8} - \frac{1}{x + 8} \cdot \frac{x + h + 8}{x + h + 8}}{h}$

Write each expression with a common denominator of $(x + h + 8) (x + 8)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x + h + 8}$ by $\frac{x + 8}{x + 8}$ .

$\frac{\frac{x + 8}{(x + h + 8) (x + 8)} - \frac{1}{x + 8} \cdot \frac{x + h + 8}{x + h + 8}}{h}$

Multiply $\frac{1}{x + 8}$ by $\frac{x + h + 8}{x + h + 8}$ .

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

Reorder the factors of $(x + h + 8) (x + 8)$ .

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

Combine the numerators over the common denominator.

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

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

Apply the distributive property.

$\frac{\frac{x + 8 - x - h - 1 \cdot 8}{(x + 8) (x + h + 8)}}{h}$

Multiply $- 1$ by $8$ .

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

Subtract $x$ from $x$ .

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

Add $0$ and $8$ .

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

Subtract $8$ from $8$ .

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

Add $- h$ and $0$ .

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

$- \frac{h}{(x + 8) (x + h + 8)} \cdot \frac{1}{h}$

Cancel the common factor of $h$ .

Move the leading negative in $- \frac{h}{(x + 8) (x + h + 8)}$ into the numerator.

$\frac{- h}{(x + 8) (x + h + 8)} \cdot \frac{1}{h}$

Factor $h$ out of $- h$ .

$\frac{h \cdot - 1}{(x + 8) (x + h + 8)} \cdot \frac{1}{h}$

Cancel the common factor.

$\frac{h \cdot - 1}{(x + 8) (x + h + 8)} \cdot \frac{1}{h}$

Rewrite the expression.

$\frac{- 1}{(x + 8) (x + h + 8)}$

Move the negative in front of the fraction.

$- \frac{1}{(x + 8) (x + h + 8)}$

Step 5

Find the Difference Quotient f(x)=1/(x+8)

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