We give an algebraic construction of the positive intersection products of pseudo-effective classes first introduced in [BDPP], and use them to prove that the volume function on the Néron-Severi space of a projective variety is C 1-differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes introduced in [ELMNP3, Ta]. We then apply our dif-ferentiability 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.