Ratio of differentials to partial derivatives of two (implicit?) functions

Hi.

So, given and

Show that

I think it makes sense (I can "see" why the equality holds) but can't think of a way to derive it. To begin, I wrote as , assuming each argument was an implicit function of the others. Similarly,

I feel like some version of the chain rule would be useful ( does the cross product of the gradients have some relationship to differential??), and that the above equality could be easier to derive if written as . I seem to have trouble taking the gradient,or any partial derivatives of implicit functions when using the chain rule... ie, is the correct derivative of f w.r.t x?

Any help appreciated, thanks.