The recent surge in algorithmic differentiation through the massive use of TensorFlow and PyTorch "autodiff" has democratized "computerized differentiation" for a broad spectrum of applications and solvers. Motivated by the challenges of nonsmoothness (such as thresholding, constraints, and ReLU) and the need to adjust parameters in various contexts directly via these solvers, we have devised tools for nonsmooth differentiation compatible with autodiff. We have in particular developed a nonsmooth implicit function calculus, aiming to provide robust guarantees for prevalent differentiation practices. We will discuss applications of these findings through the differentiation of algorithms and equations.