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?

Just so we're clear, we have $f(x) = c-(1+x+x^2+x^3+\cdots+x^n) = c-\frac{1-x^{n+1}}{1-x}$? — Brian Tung, Aug 13 '19 at 00:32
Do you also want to calculate what those roots actually are? — Lucio Tanzini, Aug 13 '19 at 00:34
@BrianTung That only holds for $ x \neq 1 $, but I need the whole domain. — , Aug 13 '19 at 17:18
@LucioTanzini Yeah, I need to roots too, but only the real roots — , Aug 13 '19 at 17:20

score 0 · Answer 1 · answered Aug 13 '19 at 00:38

0

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

answered Aug 13 '19 at 00:38

Lucio Tanzini

1 Answers1