Partial and Total Differentiation

Abstract

The notion of derivative of a function of one-variable does not really have a solitary analogue for functions of several variables. Indeed, for a function of two (or more) variables, there is a plethora of derivatives depending on whether we choose to become partial to one of the variables, or opt to move about in a specific direction, or prefer to take the total picture in consideration. The first two viewpoints lead to the notions of partial derivatives and directional derivatives, while the last leads to a somewhat more abstract notion of differentiability and, in turn, to the notion of total derivative.We define partial and directional derivatives in Section 3.1, and prove a number of basic properties including two distinct analogues of the mean value theorem and a version of Taylor’s theorem using higher-order directional derivatives. In Section 3.2, we study the notion of differentiability and prove the classical version of the Implicit Function Theorem. It may be remarked that those wishing to bypass the abstract notion of differentiability can always replace it, wherever invoked, by a slightly stronger but more pragmatic condition on the existence and continuity of partial derivatives. (See Proposition 3.33.) These readers can, therefore, skip all of Section 3.2 except perhaps the classical version of the Implicit Function Theorem. Some key results regarding differentiable functions of two variables such as the classical version of Taylor’s theorem and the chain rule are discussed in Section 3.3. Next, in Section 3.4, we revisit the notions of monotonicity, bimonotonicity, convexity, and concavity introduced in Chapter 1, and relate these to partial derivatives. Finally, in Section 3.5, we briefly outline how some of the results discussed in previous sections extend to functions of three variables, and also discuss the notions of tangent plane and normal line, which can be better understood in the context of surfaces defined (implicitly) by functions of three variables.