Definition:Formal Derivative of Polynomial

Definition

Let $R$ be a ring.

Let $R \left[{X}\right]$ be the polynomial ring over $R$.

Let $f = a_0 + a_1 X + \cdots + a_n X^n \in R \left[{X}\right]$.

The formal derivative $f'$ of $f$ is the polynomial:

$f' \left({X}\right) = a_1 + 2 a_2 X + \cdots + n a_n X^{n-1}$