Question: How do I formally differentiate?

I have worked non-stop every day for the past 3 months on a massively difficult generalization of Bell polynomials.  Sufficiently complex examples on Maple might help me to visualize the general case. One published math paper in particular has helped me tremendously.

"Generalization of the Formula of Faa Di Bruno for a Composite Function with a Vector Argument" by Rumen L. Mishkov in the Volume 24 No 7 (2000) issue of the International Journal of Mathematics and Mathematical Sciences has generalized the problem in one necessary direction, but I need one further generalization of Mishkov's Theorem.

The general problem: given G(x(1)(t(1),...,t(s)), x(2)(t(1),...,t(s),...,x(r)(t(1),...,t(s))

determine the formula for a given mixed partial derivative of G with respect to the t's in terms of the various products of powers of the mixed partial derivatives of the x's in terms of the t's.

Mishkov gives the formula for s=1, i.e. just a single "terminal" variable

So, in Maple, perhaps if I did the case r=2, s=2  G=G(x(t,u),y(t,u)), and allow me to formally compute various partial derivatives of G with respect to t and u, I might be able to see the general pattern.

Is this possible? How could I do that?

e.g. d^5 G / d^2 t d^3 u  = polynomial in d^(i+j) G / d^i x d^j y and d^(k+l) x/ d^k t d^l u

and

One other paper made a different generalization, but gave only a bunch of recursion, no final formula like Mishkov's paper does.

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