  To provide an analytic verification of the nature of phase transition at the
critical point of $P-V$ criticality, the original expressions of Ehrenfest
equations have been introduced directly. By treating the cosmological constant
and its conjugate quantity as thermodynamic pressure and volume respectively,
we carry out analytical check of classical Ehrenfest equations. To show that
our approach is universal, we investigate not only higher-dimensional charged
AdS black holes, but also rotating AdS black holes. Not only are the examples
of Einstein gravity shown, but also the example of modified gravity is
presented for Gauss-Bonnet AdS black holes. The specific heat at constant
pressure $C_P$, the volume expansion coefficient $\alpha$ and the isothermal
compressibility coefficient $\kappa_T$ are found to diverge exactly at the
critical point. It has been verified that both Ehrenfest equations hold at the
critical point of $P-V$ criticality in the extended phase spaces of AdS black
holes. So the nature of the critical point of $P-V$ criticality of AdS black
holes has been demonstrated analytically to be a second-order phase transition.
These results are consistent with the nature of liquid-gas phase transition at
the critical point. In this sense, our research would deepen the understanding
of the relations of AdS black holes and liquid-gas systems. Moreover, our
successful approaches to introduce the original expressions of Erhenfest
equations directly into black hole phase transition research demonstrate again
that black hole thermodynamics is closely related to classical thermodynamics,
which allows us to borrow techniques from classical thermodynamics to
investigate the thermodynamics of black holes.
