ABSTRACT
A great deal is known now about the behavior of ordinary derivatives of real functions. As research has progressed to place this analysis in other settings or employ generalized versions of ordinary differentiation it has been natural to seek this same behavior in the more generalized settings. The program that has been frequently followed is to investigate the extent to which the various generalized derivatives share or do not share in the host of properties possessed by ordinary derivatives. This chapter follows this program as it applies to the symmetric and the approximate symmetric derivatives. The chapter covers a fairly complete understanding of the exact Baire classification of most of these kinds of derivates. It presents the success of the Denjoy-Young-Saks theorem in classifying the relations that hold among the four Dini derivatives has prompted a similar study in almost every other setting where the ideas make sense for the various symmetric derivates.
