Based on the projector introduced in recent Euler semi-implicit methods, the main contribution of this paper is the reduction of the effect of noise on the sliding surface of the discretized homogeneous differentiator. For that, an adaptation law according to the magnitude of the high frequency noise is proposed to modify the behavior of the projector when the discretized differentiator remains on the sliding surface. In presence of noise, the differentiator will have an asymptotic convergence but will be less-sensing noise. Otherwise, when the differentiator is not affected by noise on the sliding surface, its convergence will be in finite-time. Experimental results are shown in order to highlight the well founded of the contribution.