Abstract
The aim of this paper is to discuss the commutativity of a Banach algebra A via its derivations. In particular, we prove that if A is a unital prime Banach algebra and A has a nonzero continuous linear derivation d : A -> A such that either d((xy)(m)) - x(m)y(m) or d((xy)(m)) - y(m)x(m) is in the centre of A for an integer m = m(x, y) and sufficiently many x, y, then A is commutative. We give examples to illustrate the scope of the main results and show that the hypotheses are not superfluous.