If you are using implicit differentiation, that is different from partial differentiation, and you need a bit more:

And you already have the other formula:

Now, you can set these equal to each other, but you won't get .

Edit: But .

Differentiating we get

Solving for , we get

Now, you can plug that in:

Now, setting those equal, you can solve for .

Edit 2: But, setting those equal just yields the identity .