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Published online by Cambridge University Press: 03 November 2016
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.
The following article was written before I saw the suggestions of Mr. J. Katz in the December issue of the Mathematical Gazette. With this article I, frankly, do not agree. The approach is on wrong lines: the connection of logarithms with the hyperbola is quite a secondary consideration; the use of the hyperbola to obtain an arithmetical approximation to e, without any further knowledge of it, is not conducive to a correct idea of the exponential; and the fact that e is lim (1 + 1/n)n is immaterial to the calculus course.
What is required is (i) the direct association of logarithmic and exponential functions; (ii) some attempt to show their continuity, the method being used which is suitable to the particular class taught, (iii) the demonstration that e is some number which can be used as a base of a set of logarithms; (iv) the evaluation of the integrals of ax and 1/x in terms of exponentials and logarithms.
* The following article was written before I saw the suggestions of Mr. J. Katz in the December issue of the Mathematical Gazette. With this article I, frankly, do not agree. The approach is on wrong lines: the connection of logarithms with the hyperbola is quite a secondary consideration; the use of the hyperbola to obtain an arithmetical approximation to e, without any further knowledge of it, is not conducive to a correct idea of the exponential; and the fact that e is lim (1 + 1/n) n is immaterial to the calculus course.
What is required is (i) the direct association of logarithmic and exponential functions; (ii) some attempt to show their continuity, the method being used which is suitable to the particular class taught, (iii) the demonstration that e is some number which can be used as a base of a set of logarithms; (iv) the evaluation of the integrals of ax and 1/x in terms of exponentials and logarithms.