I started solving the first few derivatives to see if I can find a pattern but I can't find it.
I tried to do something with general Leibniz rule but I don't really know if it's correct. $f'(x)=\frac{2x}{(1-x^2)^2}=2xf(x)^2$
$f^{(n)}(x)=[\text{Leibniz}]=2x(f(x)^2)^{(n-1)} + 2(n-1)(f(x)^2)^{(n-2)}$
Is this correct? What can I do next or is there another way of solving it?