find the difference quotient and simplify. f(x)=1/(x - 7) the difference quotient of f(x) is

Explanation:

Step1: Recall the difference - quotient formula

The difference - quotient formula is $\frac{f(x + h)-f(x)}{h}$, where $f(x)=\frac{1}{x - 7}$ and $f(x + h)=\frac{1}{(x + h)-7}=\frac{1}{x+h - 7}$.

Step2: Substitute $f(x + h)$ and $f(x)$ into the formula

$\frac{f(x + h)-f(x)}{h}=\frac{\frac{1}{x + h-7}-\frac{1}{x - 7}}{h}$.

Step3: Find a common denominator for the numerator

The common denominator of the numerator is $(x + h - 7)(x - 7)$. So, $\frac{1}{x + h-7}-\frac{1}{x - 7}=\frac{(x - 7)-(x + h - 7)}{(x + h - 7)(x - 7)}=\frac{x-7 - x - h + 7}{(x + h - 7)(x - 7)}=\frac{-h}{(x + h - 7)(x - 7)}$.

Step4: Divide the result of the numerator by $h$

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

Answer:

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