We introduce a new, principled approach to extend gradient-based optimization to piecewise smooth models, such as k-histograms, splines, and segmentation maps. We derive an accurate form of the weak Jacobian of such functions and show that it exhibits a block-sparse structure that can be computed implicitly and efficiently. We show that using the redesigned Jacobian leads to improved performance in applications such as denoising with piecewise polynomial regression models, datafree generative model training, and image segmentation.