This letter presents approaches that reduce the computational demand of including second-order dynamics sensitivity information into optimization algorithms for robots in contact with the environment. A full second-order differential dynamic programming (DDP) algorithm is presented where all the necessary dynamics partial derivatives are computed with the same complexity as DDP’s first-order counterpart, the iterative linear quadratic regulator (iLQR). Compared to linearized models used in iLQR, DDP more accurately represents the dynamics locally, but it is not often used since the second-order partials of the dynamics are tensorial and expensive to compute. This work illustrates how to avoid the need for computing the derivative tensor by instead leveraging reverse-mode accumulation of derivatives, extending previous work for unconstrained systems. We exploit the structure of the contact-constrained dynamics in this process. The performance of the proposed approaches is benchmarked with a simulated model of the MIT Mini Cheetah executing a bounding gait.