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 .