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(** *   Value of the derivative of the reciprocal function        *)
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Lemma derive_pt_recip_interv : forall (f g:R->R) (lb ub x:R)
(lb_lt_ub:lb < ub) (x_encad:f lb < x < f ub)
(f_incr:forall x y : R, lb <= x -> x < y -> y <= ub -> f x < f y)
(g_wf:forall x : R, f lb <= x -> x <= f ub -> lb <= g x <= ub)
(Prf:forall a : R, lb <= a <= ub -> derivable_pt f a)
(f_eq_g:forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x)
(Df_neq:derive_pt f (g x) (derivable_pt_recip_interv_prelim1 f g lb ub x
lb_lt_ub x_encad g_wf Prf) <> 0),
derive_pt g x (derivable_pt_recip_interv f g lb ub x lb_lt_ub x_encad f_eq_g
g_wf f_incr Prf Df_neq)
=
1 / (derive_pt f (g x) (Prf (g x) (derive_pt_recip_interv_prelim1_1 f g lb ub x
lb_lt_ub x_encad f_incr g_wf f_eq_g))).