 
Summary: Topics in Generalized Differentiation
J. Marshall Ash
Abstract
The course will be built around three topics:
(1) Prove the almost everywhere equivalence of the Lp
nth sym
metric quantum derivative and the Lp
Peano derivative.
(2) Develop a theory of nonlinear generalized derivatives, for ex
ample of the form anf(x + bnh)f(x + cnh).
(3) Classify which generalized derivatives of the form anf(x +
bnh) satisfy the mean value theorem.
1 Lecture 1
I will discuss three types of difference quotients. The first are the additive
linear ones. These have been around for a long time. One can see their
shadow already in the notation Leibnitz used for the dth derivative,
dd
dxd
.
For an example with d = 2, let h = dx, and consider the Schwarz generalized
