What you had is this
dqdt = Do[Print[D[q[t], t]], {t, 0, 100}]
Which does not work, since "t" becomes a number, and one can't take derivative with respect to a number. You could write
dqdt = Do[Print@Evaluate[D[q[t], t] /. t -> i], {i, 0, 100}]
But it is easier to write
lst = (D[q[t], t] /. t -> #) & /@ Range[0, 100]
Or just take the derivative outside once, and evaluate it
D[q[t], t];
lst = (% /. t -> #) & /@ Range[0, 100]