Many people who have taken a Calculus sequence of one form or another are familar with the notion that if 
which imples that if 
Fewer, might be aware of this next formula, which we get when we take the derivative with respect to x once more:
which in conjunction with the first formula gives:
This is a procedure that you can carry out indefinitely, attaining expressions for the derivative of y in terms of the partial derivatives of the implicit function. Or, you could let a computer do it for you: I wrote a bit of Mathematica Code to generate these formulas:
n = 2; Solve[Table[D[F[x, y[x]], {x, i}] == 0, {i, 1, n}], Table[D[y[x], {x, i}], {i, 1, n}]]
You can get more formulas by changing n=2 to n=3, and so on. Warning: on my machine, by n=5, there is a noticeable delay.
The second equation you have up there is pretty cool. It took me a few seconds to see exactly where it came from. Great exercise in using the product rule.
Thanks for the feedback. I updated my post to make that bit clearer.