# Changing bounds of summation after differentiating symbolic sums

Suppose, I have a function written as Taylor-Maclaurin series

f = Sum[c[n]*x^n, {n, 0, Infinity}]


Now, I wish to differentiate this expression with respect to x symbolically

D[f, x]
(* --->
Sum[n*x^(-1 + n)*c[n], {n, 0, Infinity}] *)


Resulting expression is correct except from the fact that expansion starts from n=1 now, because n=0 term vanishes after differentiation (if done explicitly).

In this case it isn't a problem because n=0 term vanishes anyway, but for more complicated computations it may be the reason for breaking further simplifications.

So, my question is:

How can I teach Mathematica to correctly change bounds of summation?

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