The problem of the stability analysis for nonlinear differential-algebraic systems is addressed using tools from classical control theory Exploiting Lyapunov Direct Method we provide linear matrix inequalities to establish stability properties of this class of systems. In addition, interpreting the differential-algebraic system as the feedback interconnection of a dynamical system and an algebraic system, a sufficient stability condition has been derived using the small-gain theorem. The proposed techniques are illustrated by means of simple examples.