In a slide it says that the closed form of $f(x)$ is $f(x)=(1-\sqrt{1-4x})/(2x)$, given that $$x f(x)^2 - f(x) + 1 = 0.$$
However, I don't see how to derive it. I tried to take derivative wrt $x$ for both sides, but failed to proceed further. $$f(x)^2 + 2xf(x)f^\prime - f^\prime = 0.$$
Any idea? Thanks!