This paper defined the $(r,s)$-differential operator of the
algebra of Laurent polynomials over the complex numbers field. Let
$\mathcal{D}_{r,s}$ be the associative algebra generated by $\{
t^{\pm 1} \}$ and the $(r,s)$-differential operator, which is called
($r,s$)-differential operators algebra. In this paper, the
derivation algebra of $\mathcal{D}_{r,s}$ and its Lie algebra
$\mathcal{D}_{r,s}^-$ were described and all the non-trivial
2-cocycles were determined.