On the Wikipedia page on the Fubini-Study metric, it gives it (for a Hilbert space) as $$\gamma(\psi,\phi)=\arccos\sqrt{\frac{\langle \psi|\phi\rangle\langle\phi|\psi\rangle}{\langle\psi|\psi\rangle\langle\phi|\phi\rangle}}$$
and says that the infitesimal form can be obtained by setting $\phi=\psi+\delta\psi$, to get $$ds^2=\frac{\langle\delta\psi|\delta\psi\rangle}{\langle\psi|\psi\rangle}-\frac{\langle\delta\psi|\psi\rangle\langle\psi|\delta\psi\rangle}{\langle\psi|\psi\rangle^2}$$
How does the derivation work? I tried to use a Taylor series approximation of arccos, which cancelled the square root, but I don't know where the rest came from.