A least-squares, continuous sensitivity analysis method was developed for transient aeroelastic gust response problems to support computationally efficient analysis and optimization of aeroelastic design problems. A key distinction between the local and total derivative forms of the sensitivity system was introduced. The continuous sensitivity equations and sensitivity boundary conditions were derived in local derivative form which was shown to be superior for several applications. The analysis and sensitivity problems were both posed in a first-order form which is amenable to a solution using the least-squares finite element method. Several example and validation problems were presented and solved, including elasticity, fluid, and fluid-structure interaction problems. Significant contributions of the research include the first sensitivity analysis of nonlinear transient gust response, a local derivative formulation for shape variation that requires parameterizing only the boundary, and statement of sufficient conditions for using nonlinear "black box" software to solve the sensitivity equations. Promising paths for future investigation were presented and discussed.