What is "subderivative"

Subderivative \Sub`de*riv"a*tive\, n. A word derived from a derivative, and not directly from the root; as, ``friendliness'' is a subderivative, being derived from ``friendly'', which is in turn a derivative from ``friend.''

n. 1 (context linguistics English) A word derived indirectly from a derivative of another 2 (context mathematics English) The slope of a line that either touches, or is below the line of a convex function

In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to functions which are not differentiable. The subdifferential of a function is set-valued. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization.

Let f:I→R be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function f(x)=|x| is nondifferentiable when x=0. However, as seen in the picture on the right, for any x in the domain of the function one can draw a line which goes through the point (x, f(x)) and which is everywhere either touching or below the graph of f. The slope of such a line is called a subderivative (because the line is under the graph of f).