We propose Differentiable Dynamic Mode Decomposition (Diff-DMD), a framework that enables gradient-based design and analysis of nonlinear dynamical systems directly through functions of DMD eigenvalues and modes [1, 2]. By embedding an adjoint-based sensitivity formulation into the standard DMD pipeline, Diff-DMD computes exact derivatives of modal quantities with respect to system parameters and data at the cost of a single forward factorization plus one adjoint solve—independent of the number of parameters. We derive sensitivities for both independently distributed and sequential snapshot constructions and further obtain data sensitivities to support robust design and adversarial data analysis. The approach yields orders-of-magnitude speedups over finite-difference approximations while maintaining accuracy. We demonstrate the method on (i) stabilizing a linear system by shifting the real parts of its eigenvalues, (ii) suppressing chaos in the Lorenz system by steering parameters until all continuous-time DMD eigenvalues have negative real parts, and (iii) increasing the oscillation frequency of a seven-dimensional yeast glycolysis model under stability and slow-decay regularization. Across these cases, Diff-DMD produces stable, accurate gradients that agree with finite differences and enable efficient optimization in high-dimensional settings, offering a scalable surrogate for design, control, and system identification based on Koopman/DMD modal structure.