We present a novel differentiable approach to quantifying and optimizing stability in robotic systems addressing an open challenge in the field of robot analysis, control, design, and optimization. Our method leverages differentiable simulation over extended time horizons to estimate a robustness metric based on the Lyapunov exponents. The proposed metric offers several properties, including a natural extension to limit cycles (commonly encountered in robotics tasks and locomotion) and independence from the trajectory path for states converging to the attractor. We showcase, with an ad-hoc JAX gradient-based optimization framework, remarkable flexibility in tackling the robustness challenge. Our approach is tested through diverse scenarios of varying complexity, encompassing high-degree-of-freedom systems and contact-rich environments. The positive outcomes across these cases highlight the potential of our method in quantifying and possibly enhancing system robustness.