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.