Constrained differentiation

In every calculus book, there is a section on the use of differentials for the approximation of increments of functions. Curiously, however, there is no such discussion in the presence of constraints, although constrained increments occur quite naturally in many situations. For example, one may want to know the possible error in the expected value of a random variable when the probabilities are measured with limited accuracy, but are constrained, of course, to sum up to one. In this paper, a very simple formula is presented for the computation of constrained differentials and the idea is then further developed into a novel alternative to the Lagrange multiplier method for constrained extrema.