This chapter intends to introduce some new n-point fractional formulas for finding an approximation for Caputo fractional differentiator. These formulas are regarded as generalizations of some n-point classical formulas for finding good approximations for the first and second classical derivatives. This would enable us to study some functions behaviors in accordance with various fractional-order values. For the purpose of achieving a high degree of precision for the acquired approaches, we employ the so-called Richardson extrapolation approach, furthermore, to demonstrate the dependability and validity of our approximations, we provide several numerical simulations via tables and graphs.