The geometric and physical properties of derivatives and integrals to an integer order are easy to describe, but fractional calculus is obviously present in modern mathematics and physics. That being said, are there a generalizations of the definitions derivatives and integrals that include these operations to arbitrary orders? To be more specific, I am not looking for a mathematical representation of one; I have seen these before. I am curious as to how a qualitative definition could describe these mathematically and/or physically with regards to the original function. I guess what I am looking for is an interpretation.