Suppose $P$ and $Q$ are polynomials and $n$ is a positive integer. It can be shown that the $n^{th}$ derivative of the rational function $f(x)=P(x)/Q(x)$ can be written in the form $f^{(n)}(x)=A_n(x)/[Q(x)]^{n+1}$ for some polynomial $A_n(x)$.
If $\displaystyle f(x)=\frac{1}{x^2+1}$, find $A_2(x)$.

$A_2(x)=$