Hello Ivette,

you can try the following:

(* Mathematica: *)
deriv = D[ArcSec[x], x];  
(* your definition: *)
someDef = 1/(x (Sqrt[x^2 - 1]));  
(* are they equivalent? *)              
eq = FullSimplify[deriv == someDef]  

which gives for "eq":

From this one can see that both expressions are the same only if x > 0 or - more general - if x is an element of the positive half-plane. "Proof":

Refine[eq, Re[x] > 0]   (* output: True *)

Henrik

EDIT:

Since the ArcSec has the form ((antisymmetric function) + Pi/2), its derivative must be symmetric. This is the case only in the result of Mathematica.