Finding the derivative of a (discrepancy) functional under minimization is an important stage in the analysis of inverse and optimization problems. This problem becomes even more complicated if the state equation involves a nonsmooth operator. The encountered difficulties can be resolved by introducing the notion of sequential operator derivative, constructed by the principle of generalized-function derivative in the sequential distribution theory. In the latter case, the equation is approximated with a family of equations that involve smooth operators. A necessary extremum condition is derived which allows one to find an approximate solution to the problem. As an example, we consider a system governed by an elliptic-type equation with nonsmooth nonlinearity.