Non integer derivative of $1/p(x)$ I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$
It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine.

The definition of a fractional derivative is
$D^{\alpha}f(x)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dx}\int_{0}^{x}\frac{f(t)}{(x-t)^{\alpha}}dt$ for $0<\alpha<1$
For higher order derivatives, compute $D^{frac(k)}f(x)=\frac{1}{\Gamma(1-frac(k))}\frac{d}{dx}\int_{0}^{x}\frac{f(t)}{(x-t)^{frac(k)}}dt$ and differentiate int(k) times normally. int(k) and frac(k) are the integer and fractional part of k, respectively.
In essence, this is your process:
$D^{k}f(x)=\frac{d^{int(k)}}{dx^{int(k)}}(\frac{1}{\Gamma(1-frac(k))}\frac{d}{dx}\int_{0}^{x}\frac{1}{p(x)(x-t)^{frac(k)}}dt)$
$=\frac{1}{\Gamma(1-frac(k))}\frac{d^{int(k)+1}}{dx^{int(k)+1}}\int_{0}^{x}\frac{1}{p(x)(x-t)^{frac(k)}}dt$
