Formulas for Differentiation of Implicit Functions

Many people who have taken a Calculus sequence of one form or another are familar with the notion that if F\left(x,y\right)=0, then by taking the derivative with respect to x, we find:

F_x+F_y y' = 0

which imples that if F_y \neq 0 then

\displaystyle y' = -\frac{F_x}{F_y}

Fewer, might be aware of this next formula, which we get when we take the derivative with respect to x once more:

F_{xx}+2 F_{xy} y' + F_{yy}\left(y'\right)^2+F_y y'' = 0

which in conjunction with the first formula gives:

\displaystyle y'' = \frac{2 F_xF_yF_{xy}-\left(F_y \right)^2F_{xx}-\left(F_x \right)^2F_{yy}}{\left(F_y\right)^3}

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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