Many people who have taken a Calculus sequence of one form or another are familar with the notion that if , then by taking the derivative with respect to x, we find:

which imples that if then

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.

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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.