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If I want to take the derivative of $ax^n$, I will get $anx^{n-1}$. If I were to take the derivative again, I get $an(n-1)x^{n-2}$.
We can generalize this for integer k easily to get the kth derivative $a\frac{n!}{(n-k)!} x ^{n-k}$. But what about for a more general k?

Does this have some name? Has it been widely studied? If so, can you show how to generalize this formula for kth derivative of $ax^n$, and explain how it works? If not, is there a good reason it is impossible?

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Here's a relevant reference: – Jonas Meyer Sep 14 '10 at 0:03
@user1881: I didn't mean my comment as a definitive and final answer. I too was hoping someone would give you a better answer. But I think that that page should be a useful source for you to at least get some ideas, and there are a number of further references that you may also find useful, in addition to any answers that are posted below. – Jonas Meyer Sep 14 '10 at 1:17
Nobody has posted this yet, so here: check out this post on John D. Cook's blog. – Tom Stephens Sep 14 '10 at 2:56
up vote 12 down vote accepted

To expand on Jonas's comment: Yes, it makes sense. For the case of the power function, one can consider


as the $\alpha$-th derivative of the power function $x^n$, where $\Gamma(z)$ is the gamma function, the generalization of the factorial to the complex plane.

In general, one has a number of definitions for so-called "fractional derivatives", or, as Spanier and Oldham prefer to call it, the "differintegral". Negative values of $\alpha$ in expressions like the one given above correspond to integration, positive values correspond to differentiation, and in general $\alpha$ can be complex.

There's a lot of things to look at (Caputo derivatives, Riemann-Liouville integrals, Grunwald-Lednikov series), and I suggest you look at the book I linked to first, and then search around the web. Have fun!

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Another book you might want to look at after Spanier and Oldham is – J. M. Sep 14 '10 at 0:46
Apparently Dover has reprinted Spanier and Oldham; if you can't get the original printing from AP, the Dover edition is quite reasonably priced... – J. M. Aug 21 '11 at 12:37
One important point worth mentioning as a comment is that the conventional integer derivatives depend only on the behavior of the function at the point where we are taking the derivative. The so called fractional derivatives do not seem to possess this kind of property. In that sense these are not generalizations of integer derivatives in a nice way. ( I came to know this after reading the link given by Jonas). – Rajesh Dachiraju Aug 21 '11 at 13:55
I always had the idea that you would take the Fourier Transform, multiply by jw^alpha, and then transform back. (Can you tell from the jw that I learned this in Electrical Engineering?) I wonder if this definition is the same as the gamma function method? – Marty Green Aug 21 '11 at 14:00
@Rajesh: Right; in particular if $\alpha$ isn't a nonnegative integer, a full determination of the differintegral requires the specification of a "lower limit" $a$, so the full notation ought to be something like ${}_a D_x^{(\alpha)} f(x)$... – J. M. Aug 21 '11 at 14:02

Here's a blog post that motivates a definition of fractional derivatives in terms of Fourier transforms.

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