Answer
$f'(x)=-\frac{2}{3}$
Work Step by Step
To take the derivative of a function using the limit process, plug into the equation $f'(x)=\lim\limits_{h \to 0}\frac{f(x+h)-f(x)}{h}$ and simplify:
$f'(x)=\lim\limits_{h \to 0}\frac{(5-\frac{2}{3}(x+h))-(5-\frac{2}{3}x)}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{5-\frac{2}{3}h-\frac{2}{3}x-5+\frac{2}{3}x}{h}$
$f'(x)=\lim\limits_{h \to 0}\frac{-\frac{2}{3}h}{h}$
$f'(x)=-\frac{2}{3}$
