half-derivative of $x^2$

Solution 1:

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

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