How do you determine the formal derivative of polynomials with coefficients in a quotient ring $GF(2)[x]/p(x)$?
I am comfortable with calculating formal derivatives of polynomials with coefficients in a prime finite field, for example if $f(x) = 3x^4 + 6x^2$ in $GF(7)[x]$ then $f'(x) = 12x^3 + 12x = 5x^3 + 5x$.
But I'm confused what to do with, for example, $f(y) = ay^4 + by^2$ where, for example, $a = x^2 + 1$ and $b = x^2 + x$ in some finite field isomorphic to $GF(2^3)$. Presumably it has something to do with $f'(y) = 4ay^3 + 2by$... but do I take $4 \bmod 2$ and $2 \bmod 2$ to determine $f'(y) = 0$? Or do I work in $\bmod 8$? Or is this not defined?