In a paper which I submitted to a peer-reviewed math journal in April 2010, I proved a formula for the n-th derivative $\frac{d^n z}{dw^n}$ in terms of (as a polynomial over the integers) of the partial derivatives of a given implicit function, $G(z,w)=0$, with respect to $z$ and $w$ (and negative integer powers of the "separant", $G_z$, the first partial derivative of $G$ with respect to $z$).

This is classic first-semester calculus homework exercise: to compute $\frac{d^n z}{dw^n}$ for n=1 and 2, namely,

$\frac{dz}{dw} = - \frac{G_w}{G_z}$

$\frac{d^2 z}{dw^2} = - \frac{G_{zz}G_{ww}}{G_{zzz}} + 2\frac{G_{zw}G_w}{G_{zz}} - \frac{G_{ww}}{G_z}$

I did so not knowing whether any one had proved the general formula first, because I am busy building on, generalizing, and using this result for other things, including chemical processing.

I have since proved the partial differential generalization of this implicit differentiation formula: i.e. given $G(z,w_1,...,w_N)=0$, compute

$\frac{d^{({u_1}+...+u_N})}{{dw_1}^{u_1}\cdots {dw_N}^{u_N}}z$ as a Laurent polynomial over the integers of the partial derivatives of $G$ with respect to $w_1,...,w_N$, and $z$

I do not have access to most peer-reviewed journals. I have had to make do with Google searches, Wolfram Research's MathWorld, online searches through my county library, and the help from one mathematician friend who has sent me related papers.

Most of the papers my friend sent me concern the Faa da Bruno formula (FdBF) and its generalizations, and the Lagrange Inversion Formula (LIF). Both the FdBF and LIF are very closely related to what I am doing, but they can not be trivially applied to get (my) general formula. (I tried... for about 8 months.) I have studied G.P. Egorychev's book: "Integral Representations of Combinatorial Sums" intensely, especially the back, with the multivariable generalizations of the LIF.

No, I am not in school. This is not a homework problem. This is "free-lance" research. I am not asking for a solution to the problem (as I already solved it independently).

I simply want to know yes or no whether someone has already done this. And, if so, where.

Thank you for the responses, in particular to go to arxiv.org, which I forgot, since I had submitted 2 papers there myself.