We consider a proper lower semicontinuous function f on a Banach space X with lambda = inf{f(x) : is an element of X} > -infinity. Let alpha greater than or equal to lambda and S(alpha) = {x is an element of X : f(x) less than or equal to alpha}. We define the lower derivative of f at the set S(alpha) by [GRAPHICS] where x. Sa can be interpreted in various ways. We show that, when f is convex and alpha = lambda, it is equal to the largest weak sharp minima constant. In terms of these derivatives and subdifferentials, we present several characterizations for convex f to have global weak sharp minima. Some of these results are also shown to be valid for nonconvex f. As applications, we give error bound results for abstract linear inequality systems.