  In Variational Analysis, VU-theory provides a set of tools that is helpful
for understanding and exploiting the structure of nonsmooth functions. The
theory takes advantage of the fact that at any point, the space can be
separated into two orthogonal subspaces: one that describes the direction of
nonsmoothness of the function, and the other on which the function behaves
smoothly and has a gradient. For a composite function, this work establishes a
chain rule that facilitates the computation of such gradients and characterizes
the smooth subspace under reasonable conditions. From the chain rule presented,
formulas for the separation, smooth perturbation and sum of functions are
provided. Several nonsmooth examples are explored, including norm functions,
max-of-quadratic functions and LASSO-type regularizations.
