Find the generating function for $c_n = na_{n-1}$

Question

Let $A(x)$ be the generating function of the sequence $(a_n)$. What is the generating function of the sequence: $$ c_n = na_{n-1} $$

In principle I know these rules:

$$ GF(a_{n-1}) = xA(x)\\ GF(na_n) = xA'(x) $$

I am not sure how to combine these two though, my idea would have been to basically first compute the generating function for $a_{n-1}$ and then simply apply the second rule (i.e. differentiate the first line) to get something like this:

$$ C(x) = x*(xA(x))' = x(A(x) + xA'(x)) $$

Can someone tell me if that is a valid approach or otherwise point me in the right direction?

Answer 1

\begin{align} \sum_{n \ge 1} n a_{n-1} x^n &= \sum_{n \ge 1} (n-1) a_{n-1} x^n + \sum_{n \ge 1} a_{n-1} x^n \\ &= x \sum_{n \ge 1} (n-1) a_{n-1} x^{n-1} + x \sum_{n \ge 1} a_{n-1} x^{n-1} \\ &= x \sum_{n \ge 0} n a_n x^n + x \sum_{n \ge 0} a_n x^n \\ &= x^2 A'(x) + x A(x) \end{align}