(** ** Definition and derivative of power function with natural number exponent *)
Definition pow_fct (n:nat) (y:R) : R := y ^ n.

(* Specifications: *)
Lemma derivable_pt_lim_pow_pos :
forall (x:R) (n:nat),
(0 < n)%nat -> derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).
Lemma derivable_pt_lim_pow :
forall (x:R) (n:nat),
derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n).
Lemma derivable_pt_pow :
forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x.
Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n).
Lemma derive_pt_pow :
forall (n:nat) (x:R),
derive_pt (fun y:R => y ^ n) x (derivable_pt_pow n x) = INR n * x ^ pred n.