Explanation

The map D is nothing but the usual derivative of a polynomial with respect to X.

It is a linear map on the vector space R[X] over R.

If R has characteristic 0, then Ker(D) = R, the 1 dimensional subspace of constants.

If R has characteristic p, then Ker(D) = R[X^p], the subalgebra of all polynomials in X^p.

The product rule D(fg) = D(f)g + fD(g) follows from linearity and the special case f = Xⁱ, g = X^j:

D(XⁱX^j) = (i+j)X^i+j-1 = iX^i-1X^j + XⁱjX^j-1 = D(f)g + fD(g).