  We derive several closed-form expressions for the fidelity
susceptibility~(FS) of the anisotropic $XY$ model in the transverse field. The
basic idea lies in a partial fraction expansion of the expression so that all
the terms are related to a simple fraction or its derivative. The critical
points of the model are reiterated by the FS, demonstrating its validity for
characterizing the phase transitions. Moreover, the critical exponents $\nu$
associated with the correlation length in both critical regions are
successfully extracted by the standard finite-size scaling analysis.
