  Consider a polarized complex manifold (X,L) and a ray of positive metrics on
L defined by a positive metric on a test configuration for (X,L). For most of
the common functionals in K\"ahler geometry, we prove that the slope at
infinity along the ray is given by evaluating the non-Archimedean version of
the functional (as defined in our earlier paper) at the non-Archimedean metric
on L defined by the test configuration. Using this asymptotic result, we show
that coercivity of the Mabuchi functional implies uniform K-stability.
