  Given a von Neumann algebra $M$ with a faithful normal semi-finite trace
$\tau,$ we consider the non commutative Arens algebra $L^{\omega}(M,
\tau)=\bigcap\limits_{p\geq1}L^{p}(M, \tau)$ and the related algebras
$L^{\omega}_2(M, \tau)=\bigcap\limits_{p\geq2}L^{p}(M, \tau)$ and
$M+L^{\omega}_2(M, \tau)$ which are proved to be complete metrizable locally
convex *-algebras. The main purpose of the present paper is to prove that any
derivation of the algebra $M+L^{\omega}_2(M, \tau)$ is inner and all
derivations of the algebras $L^{\omega}(M,\tau)$ and $L^{\omega}_2(M, \tau)$
are spatial and implemented by elements of $M+L^{\omega}_2(M, \tau).$
