Optimization of nonlinear (or linear state-dependent) dynamic systems often requires system simulation. In many cases the associated state derivative evaluations are computationally expensive, resulting in simulations that are significantly slower than real-time. This makes the use of optimization techniques in the design of such systems impractical. Optimization of these systems is particularly challenging in cases where control and physical systems are designed simultaneously. In this article, an efficient two-loop method, based on surrogate modeling, is proposed for solving dynamic system design problems with computationally expensive derivative functions. A surrogate model is constructed for only the derivative function instead of the complete system analysis, as is the case in previous studies. This approach addresses the most expensive element of system analysis (i.e., the derivative function), while limiting surrogate model complexity. Simulation is performed based on the surrogate derivative functions, preserving the nature of the dynamic system, and improving estimation accuracy. The inner loop solves the system optimization problem for a given derivative function surrogate model, and the outer loop updates the surrogate model based on optimization results. This solution approach presents unique challenges. For example, the surrogate model approximates derivative functions that depend on both design and state variables. As a result, the method must not only ensure accuracy of the surrogate model near the optimal design point in the design space, but also the accuracy of the model in the state space near the state trajectory that corresponds to the optimal design. This method is demonstrated using two simple design examples, followed by a wind turbine design problem. In the last example, system dynamics are modeled using a linear state-dependent model where updating the system matrix based on state and design variable changes is computationally expensive.