Answer
$r'=\frac{2θcosθ+2sinθ}{2(2θsinθ)^{1/2}}$
Work Step by Step
Rewrite the equation: $r=(2θsinθ)^{1/2}$
Take the derivative the equation using a Trigonometric derivative and Chain Rule:
$r'=\frac{1}{2}(2θsinθ)^{-1/2}\times(2θcosθ+2sinθ)$
$=\frac{2θcosθ+2sinθ}{2(2θsinθ)^{1/2}}$
