Section PolyOverRing.
Variables (S : {pred R}) (ringS : subringPred S) (kS : keyed_pred ringS).
Canonical polyOver_smulrPred := SmulrPred (polyOver_mulr_closed kS).
Canonical polyOver_subringPred := SubringPred (polyOver_mulr_closed kS).
Fact deriv_is_linear : linear deriv.
Canonical deriv_additive := Additive deriv_is_linear.
Canonical deriv_linear := Linear deriv_is_linear.
Definition derivE := Eval lazy beta delta [morphism_2 morphism_1] in
(derivZ, deriv_mulC, derivC, derivX, derivMXaddC, derivXsubC, derivM, derivB,
derivD, derivN, derivXn, derivM, derivMn).
(* Iterated derivative. *)
Definition derivn n p := iter n deriv p.
Local Notation "a ^` ( n )" := (derivn n a) : ring_scope.
Fact derivn_is_linear n : linear (derivn n).
Fact nderivn_is_linear n : linear (nderivn n).
Canonical nderivn_additive n := Additive(nderivn_is_linear n).
Canonical nderivn_linear n := Linear (nderivn_is_linear n).

Lemma nderiv_taylor_wide n p x h :
GRing.comm x h -> size p <= n ->
p.[x + h] = \sum_(i < n) p^`N(i).[x] * h ^+ i.