  Both the incompressibility \Ka of a finite nucleus of mass A and that
($K_{\infty}$) of infinite nuclear matter are fundamentally important for many
critical issues in nuclear physics and astrophysics. While some consensus has
been reached about the $K_{\infty}$, accurate theoretical predictions and
experimental extractions of $K_{\tau}$ characterizing the isospin dependence of
\Ka have been very difficult. We propose a differential approach to extract the
\Kt and \Ki independently from the \Ka data of any two nuclei in a given
isotope chain. Applying this novel method to the \Ka data from giant monopole
resonances in even-even Sn, Cd, Ca, Mo and Zr isotopes taken by U. Garg {\it et
al.} at the Research Center for Nuclear Physics (RCNP), Osaka University,
Japan, we find that the $^{106}$Cd-$^{116}$Cd and $^{112}$Sn-$^{124}$Sn pairs
having the largest differences in isospin asymmetries in their respective
isotope chains measured so far provide consistently the most accurate
up-to-date \Kt value of $K_{\tau}=-616\pm 59$ MeV and $K_{\tau}=-623\pm 86$
MeV, respectively, largely independent of the remaining uncertainties of the
surface and Coulomb terms in expanding the $K_{\rm A}$.
