(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(*********************************************************)
(**     Definition of the derivative,continuity          *)
(*                                                       *)
(*********************************************************)
Local Open Scope R_scope.
(*********)
Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x.
(*********)
Definition continue_in (f:R -> R) (D:R -> Prop) (x0:R) : Prop :=
limit1_in f (D_x D x0) (f x0) x0.
(*********)
Definition D_in (f d:R -> R) (D:R -> Prop) (x0:R) : Prop :=
limit1_in (fun x:R => (f x - f x0) / (x - x0)) (D_x D x0) (d x0) x0.

(*********)
Lemma D_pow_n :
forall (n:nat) (D:R -> Prop) (x0:R) (expr dexpr:R -> R),
D_in expr dexpr D x0 ->
D_in (fun x:R => expr x ^ n)
(fun x:R => INR n * expr x ^ (n - 1) * dexpr x) (
Dgf D D expr) x0.