(** * Definitions on [R] *)

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

Lemma is_derive_filterdiff (f : R -> R -> R) (x y : R) (dfx : R -> R -> R) (dfy : R) : locally (x,y) (fun u : R * R => is_derive (fun z => f z (snd u)) (fst u) (dfx (fst u) (snd u))) -> is_derive (fun z : R => f x z) y dfy -> continuous (fun u : R * R => dfx (fst u) (snd u)) (x,y) -> filterdiff (fun u : R * R => f (fst u) (snd u)) (locally (x,y)) (fun u : R * R => plus (scal (fst u) (dfx x y)) (scal (snd u) dfy)).