In the equation f(x-y)=f(x)/f(y) there are two independent variables, x and y. It does not make sense to talk about dy/dx because x and y are independent of each other.

What you need to do is to differentiate partially with respect to x and y. If you differentiate partially with respect to x then you find that . If you differentiate partially with respect to y then you find that . That gives you enough information to solve the problem.