Robot design optimization, imitation learning and system identification share a common problem which requires optimization over robot or task parameters at the same time as optimizing the robot motion. To solve these problems, we can use differentiable optimal control for which the gradients of the robot's motion with respect to the parameters are required. We propose a method to efficiently compute these gradients analytically via the differential dynamic programming (DDP) algorithm using sensitivity analysis (SA). We show that we must include second-order dynamics terms when computing the gradients. However, we do not need to include them when computing the motion. We validate our approach on the pendulum and double pendulum systems. Furthermore, we compare against using the derivatives of the iterative linear quadratic regulator (iLQR), which ignores these second-order terms everywhere, on a co-design task for the Kinova arm, where we optimize the link lengths of the robot for a target reaching task. We show that optimizing using iLQR gradients diverges as ignoring the second-order dynamics affects the computation of the derivatives. Instead, optimizing using DDP gradients converges to the same optimum for a range of initial designs allowing our formulation to scale to complex systems.