(** * Definitions on [R] *)

Definition Derive (f : R -> R) (x : R) := real (Lim (fun h => (f (x+h) - f x)/h) 0).

Lemma Derive_sum_n (f : nat -> R -> R) (n : nat) (x : R) : (forall k, (k <= n)%nat -> ex_derive (f k) x) -> Derive (fun y => sum_n (fun k => f k y) n) x = sum_n (fun k => Derive (f k) x) n.