A formulation for sensitivity analysis of fully coupled time-dependent aeroelastic problems is given in this paper. Both forward sensitivity and adjoint sensitivity formulations are derived that correspond to analogues of the non-linear aeroelastic analysis problem. Both sensitivity analysis formulations make use of the same iterative disciplinary solution techniques used for analysis, and make use of an analogous coupling strategy. The information passed between fluid and structural solvers is dimensionally equivalent in all cases, enabling the use of the same data structures for analysis, forward and adjoint problems. The fully coupled adjoint formulation is then used to perform rotor blade design optimization for a four bladed HART2 rotor in hover conditions started impulsively from rest. The effect of time step size and mesh resolution on optimization results is investigated. Results indicate that good optimization results can be obtained using time steps as large as 2 degrees, and that optimizations obtained on the coarse mesh level can be used to initialize fine mesh optimization problems in order to reduce overall computational effort. Future work will focus on aeroelastic optimizations for forward flight cases.