On derivations of some classes of Leibniz algebras

In the paper we describe the derivations of complex $n$-dimensional naturally graded filiform Leibniz algebras $NG{F}_1,NG{F}_2\text{ and }NG{F}_3\mathrm{.}$ We show that the dimension of the derivation algebras of $NG{F}_1$ and $NG{F}_2$ equals $n+1$ and $n+2,$ respectively, while the dimension of the derivation algebra of $NG{F}_3$ is equal to $2n-1.$ The second part of the paper deals with the description of the derivations of complex $n$-dimensional filiform non Lie Leibniz algebras, obtained from naturally graded non Lie filiform Leibniz algebras. It is well known that this class is split into two classes denoted by $FL{b}_n$ and $SL{b}_n\mathrm{.}$ Here we found that for $L\in FL{b}_n$ we have $n-1 \leq dim
Der(L)\leq n+1$andforalgebras$L$from$SLb_n$theinequality$n-1\leq dim
Der(L) \leq n+2$ holds true.