In this  paper we  show that under certain conditions 

the range of derivations of Banach algebras is contained 

in the radical. Our results extend those obtained 

in [14] and [2].

 For example, let $A$ be  a Banach algebra over 

the complex field. Suppose there exists a derivation 

$D:A \rightarrow A$, such that $\alpha D^3 +D^2$ is 

a derivation for some  $\alpha \in \Bbb C$. 

Then $D$ maps into its radical.