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Derivative of multiargument function

If I have the assignement:
g[z_] = 0
then any time I have:
I obtain zero and this is ok.
But if I have a sequence of functions with j=1,2,...
and I give the assignment:
g[z_,1] = 0
then Mathematica doesn't seem to give zero when it finds:
as before...
does anybody know how to let Mathematica compute the derivative even
in this case ?
Thank you very much...

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