A new Definition of Fractional Derivative and Fractional Integral

In this paper, we introduce three different definitions of fractional derivatives, namely riemann-liouville derivative, caputo derivative and the new formula caputo expansion formula, and some basics properties of these derivatives are discussed. the difference between caputo and riemann – liouville formulas for the fractional derivatives also mentioned. the paper focuses on find approximate values for function derivatives, when the function order is a negative integer, illustrated by some theorems and examples.