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.