  We consider a Banach algebra $A$ with the property that, roughly speaking,
sufficiently many irreducible representations of $A$ on nontrivial Banach
spaces do not vanish on all square zero elements. The class of Banach algebras
with this property turns out to be quite large -- it includes $C^*$-algebras,
group algebras on arbitrary locally compact groups, commutative algebras,
$L(X)$ for any Banach space $X$, and various other examples. Our main result
states that every derivation of $A$ that preserves the set of quasinilpotent
elements has its range in the radical of $A$.
