#### Answer

$f'(x) = \frac{-1}{(x+2)^{2}}$

#### Work Step by Step

$f(x) = \frac{1}{x+2}$
$f'(x) = \lim\limits_{z \to x}\frac{\frac{1}{z+2} - \frac{1}{x+2}}{z-x}$
$f'(x) =\lim\limits_{z \to x}\frac{1}{z-x}\frac{x-z}{(z+2)(x+2)}$
$f'(x) =\lim\limits_{z \to x}\frac{-1}{(z+2)(x+2)}$
$f'(x) = \frac{-1}{(x+2)^{2}}$