Value of the First derivative at the roots of a polynomial

Question

For a polynomial $ f: \mathbb{R} \rightarrow \mathbb{R} $,

$$ f(x) = c - \sum_{k=0}^n x^k $$ where $ c \geq 0 $

I would like to find the value of $ f'(x) $ for all the roots $ f(x) = 0 $, for Dirac Delta composition.

I haven't been able to get very far. How should I go about doing this?

Answer 1

If $f (x)=(x-a)g (x)$ then, $f'(x)=(x-a)g'(x)+g (x) $ and so $f'(a)=g (a) $.