Section Derive.
Context {K : AbsRing} {V : NormedModule K}.
Definition is_derive (f : K -> V) (x : K) (l : V) :=
filterdiff f (locally x) (fun y : K => scal y l).
Definition ex_derive (f : K -> V) (x : K) :=
exists l : V, is_derive f x l.
End Derive.
(** * Definitions on [R] *)
Definition Derive (f : R -> R) (x : R) := real (Lim (fun h => (f (x+h) - f x)/h) 0).
(** Derive is correct *)

Lemma Derive_correct f x :
ex_derive f x -> is_derive f x (Derive f x).