Answer
$f_y(1,\frac{1}{2})=\frac{\pi}{6}+\frac{1}{\sqrt3}$.
Work Step by Step
$f(x,y)=y\arcsin{(xy)}$
$f_y=\arcsin{(xy)}+\frac{xy}{\sqrt{1-x^2y^2}}\Rightarrow f_y(1,\frac{1}{2})=\arcsin{((1)(\frac{1}{2}))}+\frac{(1)(\frac{1}{2})}{\sqrt{1-(1)^2(\frac{1}{2})^2}}=\frac{\pi}{6}+\frac{1}{\sqrt3}$
