What can I do with half-derivative? If I want to know the slope of the tangent of curve function I just have to find the derivative of this function and if I want to know the area under this curve I can integrate it (the function of course) but what can I do with half-derivative ? 
Thank you in advance 
 A: From your own question : << If I want to know the slope of the tangent of curve function I just have to find the derivative of this function and if I want to know the area under this curve I can integrate it (the function of course) but what can I do with half-derivative >>, you can expect that fractional derivative will yield to something intermediate between the derivate function and the function itself. But that is notoriously difficult of interpretation.
The bibliography on the subject is extensive. A paper for general public, such as https://fr.scribd.com/doc/14686539/The-Fractional-Derivation-La-derivation-fractionnaire will not give you a sufficient answer. Also, the relationship between fractionnal calculus and fractal geometry is especially to be considered. In an attempt to go further, more specific papers are necessary. For example, among a lot of them :
"Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation" : http://arxiv.org/pdf/math/0110241.pdf
"Geometry of fractional spaces": http://arxiv.org/pdf/1106.5787.pdf
