The analytical sensitivity analysis, i.e., the analytical first-order partial derivatives of dynamical equations, is one key to improving descent-based optimization methods for motion planning and control of robots. This paper proposes an efficient algorithm that recursively evaluates the analytic gradient of the dynamical equations of a multibody system. The theory of projective geometric algebra (PGA) is used to generate the algorithm. It provides a systemic and geometrically intuitive interpretation for the multibody system dynamics, and the resulting algorithm is highly efficient, with concise formula. The algorithm is first applied to the open-chain system and extended for the cases when kinematic loops are contained. The runtime varying with respect to the degree-of-freedom (DOF) of the system is analyzed. The results are compared with that obtained from the algorithm based on spatial vector algebra (SVA) using open-source matlab codes. A 2DOF serial robot, a 3DOF robot with a kinematic loop and the PUMA560 robot are used for the validation of the minimum-effort motion planning, and it is verified that the proposed algorithm improves the efficiency.