The Differential Calculus

Abstract

In this chapter we give an extremely compact treatment of the first fundamental part of mathematical analysis. This is the concept of the derivative of a function, computing the derivatives of various functions, which is to say, differentiating them. To define the derivative, we must slightly alter the definition of limit as given in Chapter IV. We then give the definition of the derivative simultaneously for the real and complex cases. This occupies §21. In §22, we work out the derivatives of a number of functions. We first show how to differentiate a power series, and this gives us the means of differentiating the functions e^z, cos z, and sin z, power series expansions for which were obtained in Chapter IV We then give the rules for differentiating products, quotients, and composite functions. The rule for differentiating an inverse function is derived from this last. Thus in §22 we obtain the derivatives of all of the elementary transcendental functions. Although the present chapter bears the title “The differential calculus”, we avail ourselves of the opportunity in it of defining integration as the inverse of differentiation. We define the primitive function, or indefinite integral, again for the real and complex cases simultaneously. We use Lagrange’s formula to prove the uniqueness of the primitive function up to an additive constant. Hence we must first prove Rolle’s theorem and Lagrange’s formula for real-valued functions.