Traditionally, derivatives are taught as a function that have... "Whole" transitions. Take the following example: If you have the function f(x) = x^2, we find that f'(x) = 2x, and that f''(x) = 2. In other words, it has a first and a second derivative. But what would it even mean to take a rational number as a derivative? I mean... If there is a half derivative for the function of x, you would first imagine instinctively that it's value would between x^2 and 2x, but beyond that I wouldn't know how to imagine what it would be, or what it would imply. Is there a derivative for every real number? Imaginary? Are there Partial Half derivatives? What utility do they have? By that I mean, under what circumstance could we use a half-derivative in Physics?