This article concerns robustness analysis for interconnections of two dynamical systems (described by upper semicontinuous differential inclusions) using a generalized notion of derivatives associated with locally Lipschitz Lyapunov functions obtained from a finite family of differentiable functions. We first provide sufficient conditions for input-to-state stability (ISS) for differential inclusions, using a class of non-smooth (but locally Lipschitz) candidate Lyapunov functions and the concept of Lie generalized derivative. In general our conditions are less conservative than the more common Clarke derivative based conditions. We apply our result to state-dependent switched systems, and to the interconnection of two differential inclusions. As an example, we propose an observer-based controller for certain nonlinear two-mode state-dependent switched systems.