  The classical Gauss--Lucas theorem describes the location of the critical
points of a polynomial. There is also a hyperbolic version, due to Walsh, in
which the role of polynomials is played by finite Blaschke products on the unit
disk. We consider similar phenomena for generic inner functions, as well as for
certain "locally inner" self-maps of the disk. More precisely, we look at a
unit-norm function $f\in H^\infty$ that has an angular derivative on a set of
positive measure (on the boundary) and we assume that its inner factor, $I$, is
nontrivial. Under certain conditions to be discussed, it follows that $f'$ must
also have a nontrivial inner factor, say $J$, and we study the relationship
between the boundary singularities of $I$ and $J$. Examples are furnished to
show that our sufficient conditions cannot be substantially relaxed.
