In this paper, by considering the notions of left-right and
right-left derivations of $BCI$-algebras, we generalize some
results on regular derivations and classify this derivation in
$P$-semisimple $BCI$-algebra and $BCK$-algebra. Then we make a
congruence relation, for any derivation $d$ of $X$ and defined the
concept of conjugate derivations. In  the sequel, we show that the set
of all equivalence classes of $X$ with respect to this relation
forms a $BCI$-algebra and we denote it by $X/d$. Finally, we get
some interesting result about these quotient algebras.