This technical note gives a method for calculating the first and second derivatives of a cost function with respect to switching times for systems with piecewise second-differentiable dynamics. Differential equations governing the linear and bilinear operators required for calculating the derivatives are presented. Example optimizations of linear and nonlinear systems are presented as evidence for the value of second-order optimization methods. One example converges in 33 iterations using second-order methods whereas the first-order algorithm requires over 30 000 iterations.