#### Answer

$g'(x) = \frac{1}{2\sqrt x}$

#### Work Step by Step

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