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In this study, sensitivity equations are derived for recursive Newton-Euler dynamics using Denavit-Hartenberg (DH) moving coordinates. The dynamics and sensitivity equations depend on the 3x3 DH rotation matrices. Compared to recursive Lagrangian formulation, which depends on 4x4 DH transformation matrices, the recursive Newton-Euler formulation requires less computational effort. In addition, recursive Newton-Euler dynamics explicitly calculates internal joint forces that are not available in recursive Lagrangian formulation. The proposed formulation can handle both prismatic joints and revolute joints. More importantly, analytical sensitivities provide aid to dynamic motion predictions. Three numerical examples are presented to verify the efficacy of the proposed algorithms including two time-minimization trajectory-planning problems and one gait prediction problem. The example model setup, optimization formulation, and simulation results are presented. All three examples are successfully optimized using the proposed dynamics and sensitivity equations. The predicted kinematic and kinetic profiles are partially validated with the data in the literature. Finally, the recursive Newton-Euler dynamics and sensitivity equations are programmed using C++ with the latest math library (Eigen) for developing a general purpose motion prediction software.