Definition nat_to_ring (n : nat) : K :=
  sum_n_m (fun _ => one) 1 n.

Lemma Derive_n_pow_smalli: forall i p x, (i <= p)%nat -> Derive_n (fun x : R => x ^ p) i x = INR (fact p) / INR (fact (p - i)%nat) * x ^ (p - i)%nat.