Answer
$$\frac{{dy}}{{dx}} = \frac{{\sqrt {1 - {{\left( {xy} \right)}^2}} + y\sqrt {1 - {{\left( {x - y} \right)}^2}} }}{{\sqrt {1 - {{\left( {xy} \right)}^2}} - x\sqrt {1 - {{\left( {x - y} \right)}^2}} }}$$
Work Step by Step
$$\eqalign{
& {\sin ^{ - 1}}\left( {xy} \right) = {\cos ^{ - 1}}\left( {x - y} \right) \cr
& {\text{Differentiate with respect to }}x{\text{}} \cr
& \frac{d}{{dx}}\left[ {{{\sin }^{ - 1}}\left( {xy} \right)} \right] = \frac{d}{{dx}}\left[ {{{\cos }^{ - 1}}\left( {x - y} \right)} \right] \cr
& {\text{Use the differentiation rules }}\frac{d}{{dx}}\left[ {{{\sin }^{ - 1}}u} \right]{\text{ and }}\frac{d}{{dx}}\left[ {{{\cos }^{ - 1}}u} \right] \cr
& \frac{1}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }}\frac{d}{{dx}}\left[ {xy} \right] = - \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }}\frac{d}{{dx}}\left[ {x - y} \right] \cr
& \frac{1}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }}\left( {x\frac{d}{{dx}}\left[ y \right] + y\frac{d}{{dx}}\left[ x \right]} \right) = - \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }}\frac{d}{{dx}}\left[ {x - y} \right] \cr
& {\text{Compute derivatives}} \cr
& \frac{1}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }}\left( {x\frac{{dy}}{{dx}} + y} \right) = - \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }}\left( {1 - \frac{{dy}}{{dy}}} \right) \cr
& \frac{x}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }}\frac{{dy}}{{dx}} + \frac{y}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }} = - \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }} + \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }}\frac{{dy}}{{dy}} \cr
& {\text{Rearrange the equation}} \cr
& \frac{x}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }}\frac{{dy}}{{dx}} - \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }}\frac{{dy}}{{dy}} = - \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }} - \frac{y}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }} \cr
& {\text{Solve the equation for }}\frac{{dy}}{{dx}} \cr
& \left( {\frac{x}{{\sqrt {1 - {{\left( {xy} \right)}^2}} }} - \frac{1}{{\sqrt {1 - {{\left( {x - y} \right)}^2}} }}} \right)\frac{{dy}}{{dy}} = - \left( {\frac{{\sqrt {1 - {{\left( {xy} \right)}^2}} + y\sqrt {1 - {{\left( {x - y} \right)}^2}} }}{{\sqrt {1 - {{\left( {xy} \right)}^2}} \sqrt {1 - {{\left( {x - y} \right)}^2}} }}} \right) \cr
& \left( {\frac{{x\sqrt {1 - {{\left( {x - y} \right)}^2}} - \sqrt {1 - {{\left( {xy} \right)}^2}} }}{{\sqrt {1 - {{\left( {xy} \right)}^2}} \sqrt {1 - {{\left( {x - y} \right)}^2}} }}} \right)\frac{{dy}}{{dy}} = - \left( {\frac{{\sqrt {1 - {{\left( {xy} \right)}^2}} + y\sqrt {1 - {{\left( {x - y} \right)}^2}} }}{{\sqrt {1 - {{\left( {xy} \right)}^2}} \sqrt {1 - {{\left( {x - y} \right)}^2}} }}} \right) \cr
& \left( {x\sqrt {1 - {{\left( {x - y} \right)}^2}} - \sqrt {1 - {{\left( {xy} \right)}^2}} } \right)\frac{{dy}}{{dy}} = - \left( {\sqrt {1 - {{\left( {xy} \right)}^2}} + y\sqrt {1 - {{\left( {x - y} \right)}^2}} } \right) \cr
& \frac{{dy}}{{dx}} = - \frac{{\sqrt {1 - {{\left( {xy} \right)}^2}} + y\sqrt {1 - {{\left( {x - y} \right)}^2}} }}{{x\sqrt {1 - {{\left( {x - y} \right)}^2}} - \sqrt {1 - {{\left( {xy} \right)}^2}} }} \cr
& \frac{{dy}}{{dx}} = \frac{{\sqrt {1 - {{\left( {xy} \right)}^2}} + y\sqrt {1 - {{\left( {x - y} \right)}^2}} }}{{\sqrt {1 - {{\left( {xy} \right)}^2}} - x\sqrt {1 - {{\left( {x - y} \right)}^2}} }} \cr} $$
