  In the present paper we use the expansion formula of the polylogarithm
function to construct approximations of the Caputo derivative which are related
to the midpoint approximation of the integral in the definition of the Caputo
derivative. The asymptotic expansion formula of the Riemann sum approximation
of the beta function and the first terms of the expansion formulas of the
approximations of the Caputo derivative of the power function are obtained in
the paper. The induced shifted approximations of the Gr\"unwald formula and the
approximations of the Caputo derivative studied in the first part of the paper
are constructed and applied for numerical solution of fractional differential
equations.
