Most higher order root finding methods require evaluation of a function and/or its derivatives at one or multiple points. There are cases where the derivatives of a given function are costly to compute. In this paper, higher order methods which do not require computation of any derivatives are derived. Asymptotic analysis has shown that these methods are approximations of root iterations. One of the main features of the proposed approaches is that one can develop multi-point derivative-free methods of any desired order. For lower order methods, these correspond to the Newton, and Ostrowski iterations. Several examples involving polynomials and entire functions have shown that the proposed methods can be applied to polynomial and non-polynomial equations.