  We leverage path differentiability and a recent result on nonsmooth implicit
differentiation calculus to give sufficient conditions ensuring that the
solution to a monotone inclusion problem will be path differentiable, with
formulas for computing its generalized gradient. A direct consequence of our
result is that these solutions happen to be differentiable almost everywhere.
Our approach is fully compatible with automatic differentiation and comes with
assumptions which are easy to check, roughly speaking: semialgebraicity and
strong monotonicity. We illustrate the scope of our results by considering
three fundamental composite problem settings: strongly convex problems, dual
solutions to convex minimization problems and primal-dual solutions to min-max
problems.
