Answer
$r' = \frac{1}{\sqrt x}(1-\frac{1}{x})$
Work Step by Step
$r = 2(\frac{1}{\sqrt x} +\sqrt x)$
($\theta$ is replaced with x for convinience.)
$r = 2(x^{-\frac{1}{2}}+ x^{\frac{1}{2}})$
$r' = 2(-\frac{1}{2}x^{-\frac{3}{2}} + \frac{1}{2}x^{-\frac{1}{2}})$
$r' = -x^{-\frac{3}{2}} + x^{-\frac{1}{2}}$
$r' = x^{-\frac{1}{2}}(1-x^{-1})$
$r' = \frac{1}{\sqrt x}(1-\frac{1}{x})$
