  In this paper we study differentiability properties of the map
$T\mapsto\phi(T)$, where $\phi$ is a given function in the disk-algebra and $T$
ranges over the set of contractions on Hilbert space. We obtain sharp
conditions (in terms of Besov spaces) for differentiability and existence of
higher derivatives. We also find explicit formulae for directional derivatives
(and higher derivatives) in terms of double (and multiple) operator integrals
with respect to semi-spectral measures.
