@vv I have not changed the polynomials, except the variables (a[3],a[4],a[5],a[6],a[7],a[8],a[9]) to (x,y,z,u,v,w,t) correspodingly. It is odd that even "a in J" have to be computed so long although it is a generator. As you said maybe I was lucky with the order of my unknowns and alphabatically in the new variables it must be (t,u,v,w,x,y,z)? 

When we test ideal membership does Groebner get invovled implcitely? 

Update: I have changed the order of variables alphabeticaly and it works now and gives the same result. Attached is the worksheet. So order is so much important...wow
Resolved_(2).mw
UPDATE 2: I still am not convinced that Groebner is working properly. I cannot image how "t" is a polynomial combination of the given generators "a,b,c,d,e,f,g". Is there any way to find this?

@vv Thanks, I understood the Worksheet Mode now. 

However, when I run your code above and ask "a in J" which obviosly is true since "a" is a generator of J, Maple thinks to much. Also if I ask "x in J" it is running too long. 

But with a[3],..,a[9] the answer was instant and that worries me. 

@vv Thanks for a reply. 

What I mean is that it is working for me too in terms of a[3],...,a[9] but as soon as I change the variables and use the same polynomials Maple is not giving me answer (yet) as it was giving right away before. 

I am using x,y,z,u,v,w,t instead of a[3]....a[9] below. 

Maple_Problem2.mw

I am new to Maple so I did not get your advice on input? what option should I use?

@Carl Love Thanks, but I meant I want to have different 7 derivatives (each having different tables). I tried to index them as MyD[1], MyD[2] but when I call them Maple keeps forgetting it and uses the last one used. 

@Carl Love Is it possible to have 7 of those derivatves?  I need to extend 7 linear maps and use them in one file in a way. 

@Kitonum true, somehow I am getting an error when using it with the following table on 7 variables:

restart;
R[3] := proc(P, v) local p; if P = a[3] then return 0; elif P = a[4] then return -2*a[3]; elif P = a[5] then return -4*a[4]; elif P = a[6] then return 0; elif P = a[7] then return -2*a[6]; elif P = a[8] then return -4*a[7]; elif P = a[9] then return -6*a[8]; elif type(P, `+`) then return map(R[3], P, v); elif type(P, `*`) or type(P, `^`) then if type(P, `^`) then op(2, P)*op(1, P)^(op(2, P) - 1)*R[3](op(1, P), v); else `+`(seq(`*`(({op(P)} minus {p})[])*`if`(p::numeric, 0, `if`(p in v, R[3](p, v), op(2, p)*op(1, p)^(op(2, p) - 1)*R[3](op(1, p), v))), p = {op(P)})); end if; end if; end proc;
R[3](3 + a[3], [a[3], a[4], a[5], a[6], a[7], a[8], a[9]]);
R[3](a[3], [a[3], a[4], a[5], a[6], a[7], a[8], a[9]]);
Warning, `p` is implicitly declared local to procedure `R[3]`
Warning, `p` is implicitly declared local to procedure `R[3]`
                               0

Update: sorry, I missed one line "nops(P)=1 and not (P in v) then  0 elif". It works great! Thank you!

@Kitonum in your code, I am getting an error sometimes when consant is involved like 3+x, or just constant itself. Is there a way to fix it, pease? ;)

@vv thanks a lot.

How do I declare that D(a)=0 or so that D(ax)=aD(x) so that D knows a is a constant?