Prove the identity for $u \neq 1$, which is derived by taking the derivative of a finite geometric sum

Question

Prove the identity for $u \neq 1$, which is derived by taking the derivative of a finite geometric sum :

$$ \sum_{k=1}^{n} ku^{k}= \frac{u}{(1-u)^{2}}\big[nu^{n+1}-(n+1)u^{n}+1\big] $$

I do not really need someone to answer the question for me. I just wanted to know where should I started. The only thing I know that I would use roots of unity. Thank you !

Answer 1

Start with $\sum_{k=1}^n u^k=\frac{u-u^{n+1}}{1-u}$ and take derivatives