Taking derivative of geometric sum---why does this work?

[oudeis]

A proof I'm reading says that when we have

Geometric sum = x

we can take the derivative of both sides of the equation, and they'll still be equal:

derivative(geometric sum) = derivative(x)

Why does this work? I've tried searching for it but haven't found a link to a good explanation. (Incidentally, if anyone can recommend a good calculus primer for understanding proofs in probability and statistics, that would be fantastic....)

Is it because it's a convergent infinite series, or is it something special about the geometric sum?

EDIT:
I've found a Proofwiki stub for this but it doesn't yet have a proof or explanation, just states the theorem.
https://proofwiki.org/wiki/Derivati...ries_of_Continuously_Differentiable_Functions

 

[HallsofIvy]

Could you explain more what you mean or give an example? I really am not sure what you are asking. If is generally true that is A= B then you can "do the same thing to both sides of the equation" and the result will still be true. That is, A+ 2= B+ 2, A^2= B^2, dA/dt= dB/dt, etc. That is a property of "=", not of "convergent series" or "geometric series".

(Your link does NOT say "derivative(series) = derivative(x)", it says "as long as convergence is uniform, deriviative(series)= sum of derivative of each term". Those are very different things. Are you asking about the proof of that second statement?)