The optimization of transient structural response poses significant computational challenges, as it requires time-integration schemes that remain stable and efficient over highly nonlinear and time-dependent design spaces. The Mixed Lagrangian Formalism (MLF) [1,2] addresses these challenges by recasting the evaluation of the system state at each time step as a constrained optimization problem. This formulation has demonstrated strong robustness and numerical stability in a wide range of transient nonlinear applications, including structural dynamics and coupled multi-physics systems, particularly in the presence of pronounced nonlinear mechanical behavior. However, the optimization-based nature of MLF complicates the evaluation of design sensitivities, which are essential for gradient-based response optimization. This work introduces a new sensitivity analysis strategy for MLF that enables the efficient computation of first-order derivatives of the system response with respect to design variables. The proposed approach is based on the direct differentiation of the Karush-Kuhn-Tucker (KKT) optimality conditions governing the MLF formulation at each time step, using the implicit function theorem. By differentiating the equilibrium conditions implicitly, the method provides consistent response sensitivities while preserving the robustness of the underlying time-integration scheme. The resulting framework facilitates gradient-based optimization of complex transient problems without introducing additional convergence difficulties in the time-history analysis. The effectiveness of the proposed methodology is demonstrated through an application involving the dynamic response optimization of a structural system equipped with tension-only elasto-plastic components and viscous damping devices. [1] Sivaselvan, M. V., Reinhorn, A. M. Lagrangian approach to structural collapse simulation. Journal of Engineering Mechanics, 132(8), 795-805, 2006. [2] Sivaselvan, M.V., Lavan, O., Dargush, G.F., Kurino, H., Hyodo, Y., Fukuda, R., Sato, K., Apostolakis, G. and Reinhorn, A.M.Numerical collapse simulation of large‐scale structural systems using an optimization‐based algorithm. Earthquake Engineering & Structural Dynamics, 38(5), pp.655-677, 2009.