Find the Difference Quotient f(x)=7/(x-6)

$f (x) = \frac{7}{x - 6}$

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.

$f (x + h) = \frac{7}{(x + h) - 6}$

Simplify the result.

Remove parentheses.

$f (x + h) = \frac{7}{x + h - 6}$

The final answer is $\frac{7}{x + h - 6}$ .

$\frac{7}{x + h - 6}$

Find the components of the definition.

$f (x + h) = \frac{7}{x + h - 6}$

$f (x) = \frac{7}{x - 6}$

$f (x + h) = \frac{7}{x + h - 6}$

$f (x) = \frac{7}{x - 6}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{7}{x + h - 6} - (\frac{7}{x - 6})}{h}$

Step 4

Simplify.

Simplify the numerator.

To write $\frac{7}{x + h - 6}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{\frac{7}{x + h - 6} \cdot \frac{x - 6}{x - 6} - \frac{7}{x - 6}}{h}$

To write $- \frac{7}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + h - 6}{x + h - 6}$ .

$\frac{\frac{7}{x + h - 6} \cdot \frac{x - 6}{x - 6} - \frac{7}{x - 6} \cdot \frac{x + h - 6}{x + h - 6}}{h}$

Write each expression with a common denominator of $(x + h - 6) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{x + h - 6}$ by $\frac{x - 6}{x - 6}$ .

$\frac{\frac{7 (x - 6)}{(x + h - 6) (x - 6)} - \frac{7}{x - 6} \cdot \frac{x + h - 6}{x + h - 6}}{h}$

Multiply $\frac{7}{x - 6}$ by $\frac{x + h - 6}{x + h - 6}$ .

$\frac{\frac{7 (x - 6)}{(x + h - 6) (x - 6)} - \frac{7 (x + h - 6)}{(x - 6) (x + h - 6)}}{h}$

Reorder the factors of $(x - 6) (x + h - 6)$ .

$\frac{\frac{7 (x - 6)}{(x + h - 6) (x - 6)} - \frac{7 (x + h - 6)}{(x + h - 6) (x - 6)}}{h}$

Combine the numerators over the common denominator.

$\frac{\frac{7 (x - 6) - 7 (x + h - 6)}{(x + h - 6) (x - 6)}}{h}$

Rewrite $\frac{7 (x - 6) - 7 (x + h - 6)}{(x + h - 6) (x - 6)}$ in a factored form.

Factor $7$ out of $7 (x - 6) - 7 (x + h - 6)$ .

Factor $7$ out of $- 7 (x + h - 6)$ .

$\frac{\frac{7 (x - 6) + 7 (- (x + h - 6))}{(x + h - 6) (x - 6)}}{h}$

Factor $7$ out of $7 (x - 6) + 7 (- (x + h - 6))$ .

$\frac{\frac{7 (x - 6 - (x + h - 6))}{(x + h - 6) (x - 6)}}{h}$

Apply the distributive property.

$\frac{\frac{7 (x - 6 - x - h - - 6)}{(x + h - 6) (x - 6)}}{h}$

Multiply $- 1$ by $- 6$ .

$\frac{\frac{7 (x - 6 - x - h + 6)}{(x + h - 6) (x - 6)}}{h}$

Subtract $x$ from $x$ .

$\frac{\frac{7 (0 - 6 - h + 6)}{(x + h - 6) (x - 6)}}{h}$

Subtract $6$ from $0$ .

$\frac{\frac{7 (- 6 - h + 6)}{(x + h - 6) (x - 6)}}{h}$

Add $- 6$ and $6$ .

$\frac{\frac{7 (- h + 0)}{(x + h - 6) (x - 6)}}{h}$

Add $- h$ and $0$ .

$\frac{\frac{7 \cdot - 1 h}{(x + h - 6) (x - 6)}}{h}$

Combine exponents.

Factor out negative.

$\frac{\frac{- (7 h)}{(x + h - 6) (x - 6)}}{h}$

Multiply $7$ by $- 1$ .

$\frac{\frac{- 7 h}{(x + h - 6) (x - 6)}}{h}$

Move the negative in front of the fraction.

$\frac{- \frac{7 h}{(x + h - 6) (x - 6)}}{h}$

Multiply the numerator by the reciprocal of the denominator.

$- \frac{7 h}{(x + h - 6) (x - 6)} \cdot \frac{1}{h}$

Cancel the common factor of $h$ .

Move the leading negative in $- \frac{7 h}{(x + h - 6) (x - 6)}$ into the numerator.

$\frac{- 7 h}{(x + h - 6) (x - 6)} \cdot \frac{1}{h}$

Factor $h$ out of $- 7 h$ .

$\frac{h \cdot - 7}{(x + h - 6) (x - 6)} \cdot \frac{1}{h}$

Cancel the common factor.

$\frac{h \cdot - 7}{(x + h - 6) (x - 6)} \cdot \frac{1}{h}$

Rewrite the expression.

$\frac{- 7}{(x + h - 6) (x - 6)}$

Move the negative in front of the fraction.

$- \frac{7}{(x + h - 6) (x - 6)}$

Step 5

Find the Difference Quotient f(x)=7/(x-6)

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