Fractional Derivative Implications/Meaning? I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of the function's slope, and the second derivative is representative of its concavity, is there any qualitative relationship between a 1/2 derivative and its original function? Or a 3/2 derivative with its respective function?

Your question is an interesting one. Obviously you are looking for the presentation for this particular theory. As other responders have already stated, the fractional derivative is usually defined in academic papers by a fractional integral. To find the "fractional derivative slope" of a function it is simply a matter of assigning a value to the unknown in the function representing the fractional derivative derived from the fractional derivative. It should be noted that this value will not usually equal the functions normal derivative value, and that normal value is the actual rate of increase of the function. Therefore, the line tangent to the function created by the fractional derivative is different from the whole number value derivative of the function. The different "slope" line created by the fractional derivative can be advantageous in solving certain problems, such as the vector of a particle under different forces, in that particular case, if the forces can be expressed in terms of a function and it's fractional derivative, it greatly simplifies a problem that could involve solving a system of differential equations by simply taking fractional derivatives.
Perhaps you are thinking of the definition of a derivative:
lim x-> <>X    (F(x + <>x) - F(x))/<>x
I remember reading a math text in which fractional derivatives were defined in terms of a limit instead of a fractional integral. Unfortunately I have forgotten the author and title. However, defining a fractional derivative in terms of a limit is much more advantageous than in terms of an integral since not all fractional integrals may be integrated. Once again it all depends on the theories presentation in both the case of a limit and integral definition.
