Answer
The solution is
$$f_y\left(1,\frac{1}{2}\right)=\frac{\pi+2\sqrt{3}}{6}.$$
Work Step by Step
First, find the derivative:
$$f_y=\Big(y\sin^{-1}(xy))\Big)'_y=(y)'_y\sin^{-1}(xy)+y\big(\sin^{-1}(xy)\big)'_y=\sin^{-1}(xy)+y\frac{1}{\sqrt{1-(xy)^2}}\cdot(xy)'_y=\sin^{-1}(xy)+\frac{xy}{\sqrt{1-x^2y^2}}.$$
Now calculate by direct substitution $x=1$ and $y=1/2$:
$$f_y\left(1,\frac{1}{2}\right)=\sin^{-1}\left(1\cdot\frac{1}{2}\right)+\frac{1\cdot\frac{1}{2}}{\sqrt{1-1^2\cdot\left(\frac{1}{2}\right)^2}}=\frac{\pi}{6}+\frac{\frac{1}{2}}{\sqrt{\frac{3}{4}}}=\frac{\pi}{6}+\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{\pi+2\sqrt{3}}{6}.$$
