THE DERIVATIVE

I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives.

Charles Hermite

In this chapter we abandon our more general notation involving X, D, and Y and work individually with the specific cases of interest in developing the derivative. We begin in Section 6.1 by considering functions f : D1 → R. There, we remind the reader of the traditional difference quotient definition for the derivative of a function at a point, a formulation with very practical benefits. Most notable among these is that this version of the definition allows for the computation of the derivative at a specified point. This definition also provides conceptual clarity, directly exhibiting the derivative f ′(a) to be the rate of change of f with respect to changes in its argument at the point a. Yet despite these strengths, the difference quotient derivative definition has one significant shortcoming. While it extends to the class of complex functions f : D → C, it is not generalizable to other real classes of functions f : Dk → R or f : Dk → Rp for k, p > 1. For this reason, we immediately develop two alternative versions of the derivative definition, “the !, δ version“ and “the linear approximation version.“ All three versions will be shown to be mathematically equivalent in the cases f : D1 → R and f : D → C, but the !, δ version and the linear approximation version are both generalizable to the higher-dimensional real-function cases, while the difference quotient version is not. And yet these alternate versions are not perfect either. The !, δ version is generalizable and practical in theoretical developments, but it lacks the conceptual clarity of the difference quotient version and cannot be used to actually compute the derivative of a function at a point. The linear approximation version is also generalizable and provides a conceptual clarity of its own, but it lacks the practical benefit that each of the other versions provides. Since each of the three derivative formulations is less than optimal in some way, we choose the linear approximation version as our “official“ definition for its generalizability and its conceptual strengths. However, it is important

to note that, depending on the circumstances, all three versions of the definition are useful. For this reason, it is important to develop a facility with each of them. The remaining part of Section 6.1 is devoted to establishing the many derivative related results for functions f : D1 → R that are so familiar to students of calculus. In Section 6.2, we extend many of these results to functions f : Dk → R for k > 1, pointing out differences and subtleties associated with the higher-dimensional domain, including the concept of partial derivatives. In Section 6.3 we handle the class of real functions f : Dk → Rp for k, p > 1, where we find that some new tools are required to deal with the fact that the derivative at a point in this case can be represented as a matrix of constants. Finally, in Section 6.4 we consider the case of a complex-valued function of a complex variable, f : D → C. As we learned in a previous chapter, C is geometrically similar to R2 and algebraically similar to R. In fact, we’ll find that the class of complex functions can be compared to the two classes of real functions represented by f : D1 → R and f : D2 → R2 and inherits the best of both worlds in many ways. Complex functions possess the extra algebraic structure that the field properties afford, as well as the richness of a two-dimensional geometric environment. Yet despite these similarities, we will discover that complex functions also possess many unique properties not found in any of the real counterparts to which we compare it. In the last section of the chapter, we state and prove the inverse and implicit function theorems. While they are relatively straightforward generalizations of what students already know about the existence of inverse and implicit functions from a first year calculus course, their proofs are subtle and involve much of the more advanced techniques of analysis developed thus far.