Stability is one of the most important subjects in control systems. As for the stability of nonlinear dynamical systems, Lyapunov's direct method and linearized stability analysis method have been widely used. But, finding an appropriate Lyapunov function is fairly difficult especially for complex nonlinear dynamical systems. Also it is hard to obtain the locally asymptotically stable region (RLAS) by these methods. Therefore, it is highly motivated to develop a new stability analysis method that can obtain RLAS easily. Accordingly, in this paper a new stability analysis method based on the higher ordered derivatives (HODs) of universal learning networks (ULNs) with ξ approximation and its application to a DC motor system are described. The proposed stability analysis method is carried out through two steps: Firstly, calculating the first ordered derivatives of any node of the trajectory with respect to the initial disturbances and checking if their values approach zero at time infinity or not. If they approach zero, then the trajectory is locally asymptotically stable. Secondly, obtaining RLAS where the first order terms of Taylor expansion are dominant compared to the second order terms with ξ approximation.