In Introduction to finite fields and their applications by R.Lidl, the definition of the derivative for a polynomial such that ($a_i\in GF(q)$) $$f_{(x)}=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$
is $$f_{(x)}=na_n*x^{n-1}+\cdots +a_1 $$
It is not clear what the term $na_n$ actually means. We already know that $a_n$ is a member of the field. However, it is not clear what $n$ means. It can be the polynomial representation or the power representation.
What is the correct definition?
$n a_n$ usually refers to the canonical group action of $\mathbb{Z}$ on the field you're considering, i.e. $$n a_n = \underbrace{a_n + \ldots + a_n}_{n \text{ times}}$$
EDIT: If you insist on using a polynomial representation, maybe this site can help.