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Question on two-stage optimization of polynomials: how to draw a response function?

I need to solve an optimization problem. Please offer your insights.
Suppose there are two functions (my case is more complicated than this

f(x,y)=x-y, 0<x<1, 0<y<1
g(x,y)=x*y, 0<x<1, 0<y<1

The goal is to get the maximum values for f(x,y) and g(x,y). However,
it is not simultaneous, but sequential. In the first stage, only x can
be chosen to maximize f(x,y); in the second stage only y can be chosen
to maximize g(x,y).

To solve this problem, I should start from the second stage, say, let
x varies from 0 to 1, and get each y that maximizes g(x,y). or in
another word, derive the best response function y*=v(x). Then I can
put the response function back to f(x,y) and solve for the x that
maximizes f(x,y).

An easy way is to get the reponse function is to differentiate g(x,y)
respect to y, and the set it to zero: df/dy=0, and the response
function is at hand. However, my case is too complex to get the
differentiation. Therefore I need to solve the problem numerically.

Therefore, what I want is a function or a curve that can let x varies
from 0 to 1 and see the best response value of y. Even if I can not
get a response function, I hope I can get a response curve drawn by

Can someone help me out this trouble?

Many thanks

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