We introduce JAX FDM, a differentiable solver to design mechanically efficient shapes for 3D structures, such as domes, cable nets and towers, conditioned on target architectural, fabrication and structural properties. JAX FDM solves these inverse form-finding problems by combining the force density method, differentiable sparsity and gradient-based optimization. JAX FDM can be paired with optimization and neural network libraries in the JAX ecosystem to facilitate the integration of form-finding simulations into neural networks. We showcase the features of JAX FDM in two structural design examples. JAX FDM is available as an open-source library.