Properties of Karcı's Fractional Order Derivative

The derivative concept was defined by Newton and Leipzig. After these scientific, there are many approaches about the order of derivative, since derivative defined by Newton and Leipzig considered as order of 1. Many scientists such as Caputo, Riemann, etc. defined the fractional order derivative. Karcı is one of them who defined fractional order derivative. was defined by Karcı, and it is not a linear derivative operator; it is a non-linear derivative operator. In this paper, we verified the most important properties of . has got an α parameter and this parameter can be any complex number. The properties of , which are derivative of product, derivative of quotient, the chain rule, the relationship between and complex numbers, etc., were verified in this paper. The most of these properties were not satisfied by other definitions for fractional order derivatives such as Caputo, Riemann-Lioville, Euler, etc. Khallil and his friends also defined fractional order derivative in a special case. This derivative satisfies these properties for special functions; in general, this definition also does not satisfy these properties.