Derivations of 2-subhomogeneous C*-algebras

A characterization is given of those unital, 2-subhomogeneous, Fell C*-alge bras which have only inner derivations. This proves Sproston and Strauss's conjecture from 1992. Various examples are given of phenomena which cannot occur for separable C*-algebras. In particular, an example is given of a C* -algebra with only inner derivations which has a quotient algebra admitting outer derivations. This answers a question of Akemann, Elliott, Pedersen a nd Tomiyama from 1976.