This paper investigates a new approach to studying several fractional derivatives in the distributional sense based on the products of distributions and the delta sequence with compact support. Furthermore, we consider an asymptotic expression to the fractional derivative of the delta function and show that it is the first-order approximation in the Schwartz space. At the end of paper, we provide several asymptotic formulas to more complicated fractional derivatives of distributions.

Tbilisi Mathematical Journal, Vol. 10(1) (2017), pp. 1-18