  Given an ample line bundle $L$ on a geometrically reduced projective scheme
defined over an arbitrary non-Archimedean field, we establish a
differentiability property for the relative volume of two continuous metrics on
the Berkovich analytification of $L$, extending previously known results in the
discretely valued case. As applications, we provide fundamental solutions to
certain non-Archimedean Monge--Amp\`ere equations, and generalize an
equidistribution result for Fekete points. Our main technical input comes from
determinant of cohomology and Deligne pairings.

Algebraic Geometry;Number Theory