  We give an algebraic construction of the positive products of
pseudo-effective classes first introduced by Boucksom, Demailly, Paun and
Peternell, and use them to prove that the volume function on the Neron-Severi
space of a projective variety is (once) differentiable. The differential is
expressed as a positive product; we also relate it to the restricted volumes
introduced by Ein et al and by Takayama. Then we apply our differentiability
result to prove an algebro-geometric version of the Diskant inequality in
convex geometry, allowing us to characterize the equality case of the
Khovanskii-Teissier inequalities for nef and big classes.
