Question

Find the difference quotient please

Answer 1

Given:
$f(x) = \frac{x}{x+6}$

Answers:
$f(a) = \frac{a}{a+6}$
$f(a+h) = \frac{a+h}{a+h+6}$
$\frac{f(a+h)-f(a)}{h} = \frac{6}{(a+h+6)(a+6)}$

Explanation:
$f(a+h)-f(a) = \frac{(a+h)(a+6)-a(a+h+6)}{(a+h+6)(a+h)} \\ = \frac{a^{2}+6a+ah+6h-(a^{2}+ah+6a)}{(a+h+6)(a+6)} \\= \frac{6h}{(a+h+6)(a+6)}$
$\frac{f(a+h)-f(a)}{h} = \frac{6}{(a+h+6)(a+6)}$