# half-derivative of $x^2$

I was given this problem to challenge me

$$\frac{d^{1/2}}{dx^{1/2}}x^2$$

I googled wikipedia, and tried to follow the steps shown.

I got an answer of $\frac{16\sqrt{ \pi x}}{9\pi}$ edited

2 part question.

a) is my answer correct? b) Reference request on a good paper/book on Fractional Calculus.

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Your answer is wrong - just double check wikipedia to see the correct proof. There are plenty of references listed as well. – Bombyx mori Sep 19 '12 at 3:12
See example 2 at: epublications.bond.edu.au/cgi/… – Amzoti Sep 19 '12 at 3:42

## 1 Answer

Related problem: (I), (II). Here is a formula where you can use it to find the fractional derivative of a monomial $x^n$,

$$\frac{d^q}{dx^q} x^m = \frac{\Gamma(m+1)}{\Gamma(m-q+1 )} x^{m-q}\,.$$

The above formula was derived using the Riemann-Liouville definition for fractional derivative

$$f^{(q)}(x) = \frac{1}{\Gamma(k-q)} \frac{d^k}{dx^k} \int_{a}^{x}\, (x-t)^{k-q-1}\,f(t)\,dt\>, \quad (k-1 < q < k )\,,$$ where $k=\lceil q \rceil$ is the ceiling of $q$.

See Chapter 2 in this book for details of derivation. In your case $q=\frac{1}{2}$, then the formula gives

$$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x^{2} = \frac{\Gamma(3)}{\Gamma(\frac{5}{2} )} x^{\frac{3}{2}} = \frac{8}{3 \sqrt{\pi}} x^{\frac{3}{2}}\,.$$

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@Michael: Thanks for the edit. – Mhenni Benghorbal May 30 '13 at 3:14