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We know that $\frac{d^{n}e^{x}}{dx^{n}}=e^{x}$. Can we define the $n$th derivation of $e^{x}$ which $n$ is a real number?!!!

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Short answer: YES. – glebovg Dec 15 '12 at 4:18
@PatrickDaSilva: Honestly, I said him to fix it frequently ,but nothing's appeared! – Babak S. Dec 15 '12 at 7:15
up vote 6 down vote accepted

Absolutely! You can search for "fractional derivative" to find a lot of beginners and advanced oriented material. The general theory (i.e., not only fractional derivatives of $e^x$) not only makes sense mathematically but has many applications.

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Thank you very much. – aliakbar Dec 15 '12 at 4:38
You can define what ever you want, how do you think definitions are created in the first place? someone defines them. – Ethan Dec 15 '12 at 4:52
@Ethan: I find your comment cryptic. Evidently the question was not if one can give some arbitrary definition and call it a fractional derivative just for the fun of it. I find the OP shows healthy curiosity by considering the absolutely non-trivial question of whether the ordinary notion of derivative can be extended to include non-integer derivatives. The fact that the answer is relatively well-known to be yes does not diminish from the question and falls well inline with the purpose of this Q&A site. – Ittay Weiss Dec 15 '12 at 5:32

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