Partial derivatives

[Ilya Zakharevich <ilya@math.ohio-state.edu>]

I'm thinking about adding new type "a transcendental function" to
PARI.  It is going to be a slight modification of a user variable
(undeterminate) with some additional rules of manipulation stored in
some database, so that

  q=Trans(sin,x^2);
  deriv(q,x)

may be calculated to be 2*x*Trans(cos,x^2).

But first I want to understand what happens with algebraic functions,
which *apparently* should be supported with the current implementation
as well.

Does not it look strange to you that

    ? q=Mod(z,x+y+z)
    %1 = Mod(z, x + (z + y))
    ? deriv(q,x)
    %2 = 0
    ? deriv(q,y)
    %3 = 0
    ? deriv(q,z)
    %4 = Mod(1, x + (z + y))

? Suppose that we have a modification 

  deriv(a,x,[y,y1,y2])

with the meaning "partial derivative of a wrt x when y,y1,y2 remain constant".

Then 

  deriv(q,x,[y])

might be calculated and be Mod(-1,x+(z+y)).

Any thoughts?

Ilya

P.S.  Why is it that I cannot make a damn from Chapter2's PolMods?  Is
      not it a codification of some bugs in PARI?  Why should
      y+Mod(x,x^2+1) behave any differently than Mod(x+y,x^2+1)?