Answer
$\frac{5}{9}$
Work Step by Step
Step 1: Simplify the difference quotient $\frac{f(x)-f(a)}{x-a}$.
$\frac{f(x)-f(3)}{x-3}=\frac{\frac{x^2}{x+6}-\frac{3^2}{3+6}}{x-3}=\frac{\frac{x^2}{x+6}-1}{x-3}=\frac{\frac{x^2-x-6}{x+6}}{x-3}=\frac{x^2-x-6}{(x+6)(x-3)}=\frac{(x+2)(x-3)}{(x+6)(x-3)}=\frac{x+2}{x+6}$
Step 2: Evaluate the limit $\lim\limits_{x \to a}\frac{f(x)-f(a)}{x-a}$.
$\lim\limits_{x \to 3}\frac{f(x)-f(3)}{x-3}=\lim\limits_{x \to 3}\frac{x+2}{x+6}=\frac{3+2}{3+6}=\frac{5}{9}$
Step 3: Conclude $f'(a)$.
$f'(3)=\lim\limits_{x \to 3}\frac{f(x)-f(3)}{x-3}=\frac{5}{9}$
