The equations to study the variation of the hardness and the Kohn-Sham Fukui function under the action of an external perturbation are given. Perturbative equations involving the Unsold approximation are presented. It is found that the variation of the Kohn-Sham Fukui function depends only on the knowledge of the highest occupied molecular orbital density and a mean energy difference which can be interpreted as the band gap or the finite difference approximation for the hardness. The equation for the variation of the hardness is more involved and it depends on the knowledge of the linear response function and the functional derivative of the exchange-correlation potential. However, a simplified way to calculate the variation of hardness is proposed.