  Differentiability of geometric and arithmetic volumes of Hermitian
line-bundles leads to the proof of equidistribution results on projective
varieties using the variational principle. In this article, we work in the
setting of adelic divisors on quasi-projective varieties recently introduced by
Xinyi Yuan and Shou-Wu Zhang to show that their geometric and arithmetic adelic
volume functions are differentiable on the big cone. We introduce the notions
of positive intersection products and show that the differentials are realised
as positive intersection products against integrable divisors at big points. We
show an analogue of the Fujita Approximation for restricted volumes of adelic
divisors in the geometric setting using a construction similar to those of
positive intersections and as an application of our differentiability result we
derive a slightly weaker quasi-projective analogue of the equidistribution
theorem of Berman and Boucksom which generalises the equidistribution obtained
by Yuan and Zhang for arithmetically nef adelic divisors on quasi-projective
varieties.
