# Question:Formal differentiation?

## Question:Formal differentiation?

Maple

I need to formally partial-differentiate a given arbitrary function G(z,w) with respect to two other variables z=z(s,t) w=w(s,t) and then to express the p-th-total-order partial derivatives of z with respect to s and t as a polynomial in the partial derivatives of G with respect to z and w and the partial derivatives of w with respect to s and t, divided by the partial derivative of G with respect to z raised to the power 2p+1.   The eliminate function allows me to eliminate all partial derivatives of z w.r.t. s and t of total order lower than p.

But, I first need to formally differentiate G(z(s,t),w(s,t)) with respect to s and t.

If Maple does not provide that function, then I need to write out the known formula given by Mishkov in a paper in 2000.

Like any of these formula for derivatives of compositions of functions, I would need some way of summing over tuples of nonnegative integers. But, Maple does not have that capability.

I need to be able to perform this computation a sufficient number of times until I can guess the general rational formula F for

d^p z / ds^m dt^n = F( d^q G / dz^a dw^b,   d^i w / ds^j dt^k) where p=m+n, q=a+b, and i=j+k

When z and w are functions of a single variable, t, and we set t=w, we reduce to the familiar case given by the Franz Kamber formulae in Acta Mathematica of 10 January 1946 which express  d^n z/ dw^n = F( dw/dz, d^2 w/dz^2, ..., d^n w/dz^n) / ( dw/dz)^(2n+1) where F = (-1)^n * sum from i=1 to n-1 of (-dw/dz)^(n-1-i) * (n-1-i)! * sum over the ordered (n-1)-tuple of nonnegative integers (a(1),...,a(n-1)) of product from j=1 to n-1 of (d^j w/dz^j)^a(j) / (a(j)! * (j!)^a(j))

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