In view of training increasingly complex learning architectures, we establish a nonsmooth implicit function theorem with an operational calculus. Our result applies to most practical problems (i.e., definable problems) provided that a nonsmooth form of the classical invertibility condition is fulfilled. A major feature of our formula, based on conservative Jacobians, is its compatibility with algorithmic differentiation (e.g., backpropagation). We provide several applications of our results: training deep equilibrium networks, training neural nets with conic optimization layers, hyperparameter tuning for nonsmooth Lasso-type models. To show the sharpness of our assumptions, we present numerical experiments showcasing the extremely pathological gradient dynamics one can encounter when applying implicit algorithmic differentiation without any hypothesis.