find the difference quotient and simplify. f(...

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