It looks to me as if you really have not made the leap to multiple variables. You are going to an awful lot of effort to separate x from y. You don't get to do that. The 2D definition of a derivative gives away the significant generalization that there are no longer only two ways to approach a limit, left or right. There are now infinitely many ways to approach the limit. For this reason, one must consider both simultaneously.
You must do some reading on:
Partial Derivative - Clearly, you have seen some of this. Find some more.
Parametric Representation of Curves