A first derivative test is formulated for determining the nature of extrema of a function f(x) of n real variables, subject to the constraints gi(x) = 0, i = 1,2.....r, where r<n. The test, which does not require the differentiability of f at the critical point, x = a, nor the existence of second order derivatives of gi, is compared with the second derivative test for such extrema. Examples are given which demonstrate advantages of the first derivative test, and it is shown that if the second derivative test classifies a strict local minimum or maximum for the constrained problem, then, in principle, so does the first derivative test.