This paper presents a modified s-to-z transformation of VVGS and VVG- Al-SKG rule, that is expanded for fractional powers based on the principle of continued fraction expansion (CFE). The magnitude response is observed to be improved at low frequencies as compared to the VVGS and VVG-AL-SKG operator. The paper also introduces a generalized design of a half differentiator based on the Al-Alaoui principle, specified for 3^rd and 4^th orders. The proposed solutions have been designed in MATLAB and corresponding results are analyzed by comparing them with the theoretical results of continuous-time ideal differentiators and other existing operators. Half-differentiator based on Al-Alaoui-Schneider based discretization operator has also been suggested for 3^rd and 4^th orders. The results are supported with numerical simulations that are provided to illustrate the effectiveness of the obtained theoretical results. The results reveal that the half differentiator based on Al-Alaoui Schneider operator outperforms the previous versions of operators in higher frequency ranges as well, with negligible increase in simulation time.