Section NullDerivative.
Context {V : NormedModule R_AbsRing}.
End NullDerivative.
(** * Iterated differential *)
(** ** Definition *)
Fixpoint Derive_n (f : R -> R) (n : nat) x :=
match n with
| O => f x
| S n => Derive (Derive_n f n) x
end.
Definition ex_derive_n f n x :=
match n with
| O => True
| S n => ex_derive (Derive_n f n) x
end.
Definition is_derive_n f n x l :=
match n with
| O => f x = l
| S n => is_derive (Derive_n f n) x l
end.
(** Extensionality *)
(** Constant function *)
(** ** Operations *)
(** *** Additive operators *)
(** Opposite *)
(** Addition of functions *)
(** Subtraction of functions *)
(** *** Multiplicative operators *)
(** Scalar multiplication *)
(** *** Composition *)
(** Composition with linear functions *)

Lemma Derive_n_comp_opp (f : R -> R) (n : nat) (x : R) :
locally (- x) (fun y => (forall k, (k <= n)%nat -> ex_derive_n f k y)) ->
Derive_n (fun y => f (- y)) n x  = ((-1) ^ n * Derive_n f n (-x)).