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D[x1Com,x[1,i]] will give zero, since there is no x[1,i] any more in x1Com. There is only x[1,1] and x[1,2] and so on.

One way could be

ClearAll[x1Com, x, m];
x1Com = HoldForm@Sum[(m[1, i]*x[1, i]), {i, 1, 6}]/HoldForm@Sum[m[1, i], {i, 1, 6}]

Now make a function to differentiate w.r.t. to specific x[1,i]

diff[x1Com_, x_] := D[ReleaseHold@x1Com, x];

Now use it

diff[x1Com, x[1, 1]]

diff[x1Com, x[1, 2]]

And so on.