How to calculate the derivative of following geometric series?

Question

How to calculate and what will be the derivative of following geometric series

\begin{align} f(q) = \sum_{t = 1}^{n} q^t = \frac{q(1 - q^n)}{1-q} \end{align}

what is $f'(q)$ when $n$ is bounded.

Answer 1

You can derive the right term using classic derivation formulas:

$$ f'(q) = \frac{1-(n+1)q^n + n q^{n+1} }{(1-q)^2} $$

And since the sum is finite you can derive every term of it, which at the end gives:

$$f'(q) = \sum_{t=1}^n t q^{t-1}$$