  The concept of fractional order derivative can be found in extensive range of
many different subject areas. For this reason, the concept of fractional order
derivative should be examined. After giving different methods mostly used in
engineering and scientific applications, the omissions or errors of these
methods will be discussed in this study. The mostly used methods are Euler,
Riemann-Liouville and Caputo which are fractional order derivatives. The
applications of these methods to constant and identity functions will be given
in this study. Obtained results demonstrated that all of three methods have
errors and deficiencies. In fact, the obtained results demonstrated that the
methods given as fractional order derivatives are curve fitting methods or
curve approximation methods. In this paper, we redefined fractional order
derivative (FOD) by using classical derivative definition and L Hospital
method, since classical derivative definition concluded in indefinite limit
such as 0/0. The obtained definition is same as classical derivative definition
in case of fractional order is equal to 1. This implies that definitions and
theorems in this paper sound and complete.
  After defining the FOD concept, the applications of FOD for polynomial,
exponential, trigonometric and logarithmic functions were handled in this
study. The properties of FOD for positive monotonic increasing/decreasing
functions were demonstrated in this paper. Another important point is that the
relation between derivatives and complex functions were also verified in this
paper.
