A family of nonstandard Gauss-Jacobi-Lobatto quadratures for numerical calculating integrals of the form $\int_{-1}^1 f'(x)(1-x)^\alpha{\D}x$, $\alpha>-1$, is derived and applied to approximation of the usual fractional derivative. A software implementation of such quadratures was done by the recent {\sc Mathematica} package {\tt OrthogonalPolynomials} (cf. [A.S.~Cvetkovi\'c, G.V.~Milovanovi\'c, Facta Univ. Ser. Math. Inform. {\bf19} (2004), 17--36] and [G.V.~Milovanovi\'c, A.S.~Cvetkovi\'c, Math. Balkanica {\bf26} (2012), 169--184]). Several numerical examples are presented and they show the effectiveness of the proposed approach.
