The Approach to the Differentiation and Integration of Logarithmic and Exponential Functions

Abstract

In teaching beginners the elements of the calculus, the teacher’s first real difficulty arises when he has to present the differentiation of the logarithmic and exponential functions; these being a natural outcome of an attempt to complete the rule for xn for all values of n, including the case when n=−1. He either has to make the approach as in texts on “practical mathematics,” by differentiating the series for ex, with all the tacit assumptions that have to be made; or he is dependent on non-rigorous treatment of limits, necessary to show that (ax−1)/x tends to the limiting value logea; or to find solutions of dy/dx=y by some such process as is given in Lamb’s treatise on the Calculus, without the necessary proofs of convergence and differentiability of the series assumed; or he is bound to discuss convergence and continuity fairly fully, and give such proofs as Lamb gives, or the equally severe method of Hardy’s text.