  We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the
uniformizing functions of genus zero Fuchsian groups of the first kind. Our
proof relies on differential Galois theory, monodromy of linear differential
equations, the study of algebraic and Liouvillian solutions, differential
algebraic work of Nishioka towards the Painlev\'e irreducibility of certain
Schwarzian equations, and considerable machinery from the model theory of
differentially closed fields.
  Our techniques allow for certain generalizations of the
Ax-Lindemann-Weierstrass theorem which have interesting consequences. In
particular, we apply our results to answer a question of Painlev\'e (1895). We
also answer certain cases of the Andr\'e-Pink conjecture, namely in the case of
orbits of commensurators of Fuchsian groups.
