Sensitivity analysis plays an important role in modern engineering applications where design optimization is desired. A computationally efficient sensitivity analysis scheme is presented in this paper in an effort to facilitate design optimization as it pertains to general, complex multi-rigid-body dynamic systems. Based on the underlying velocity space projection, state space formulation, and direct differentiation approach, the first-order sensitivity information can be efficiently determined in a fully recursive manner for general multi-rigid-body dynamic systems involving an arbitrary number of closed loops. The overall computational expense of this method is bilinear in the number of design variables and the number of system generalized coordinates. The solution accuracy and the computational performance are demonstrated by several numerical examples.