Abstract
Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH1(B) with coefficients in the B–B-bimodule B. If C is a tilted algebra such that B is the relation extension of C by
$E = {\rm Ext}_{C}^{2}({DC},{C})$
, then we prove that HH1(B) is isomorphic, as a vector space, to the direct sum of HH1(C) with HH1(B,E). This yields homological interpretations for results of the first and the fourth authors with M. J. Redondo.