Differentiability of the $\chi$-volume function over an adelic curve

In this article, we show a differentiability property for the $\chi$-volume function on the ample cone of adelic line bundles over an adelic curve. This result is deduced from a non-Archimedean counterpart of a diffrentiability result of Witt Nyström. As an application, we give a logarithmic equidistribution result over adelic curves.