Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an Wisconsin, he returne to Oregon State, where he spent the rest of his career. His mathematical wanerlust le to sabbaticals at Wisconsin, Michigan State, an Hamburg (on a Senior Humbolt awar), an shorter stays in other universities in Germany, Switzerlan, Englan, Italy, Russia, an Australia. His specialty is integral equations an operator approximation theory. He also has a keen interest in evising elegant proofs of important results in calculus an its applications. John Lee (jwlee@math.orst.eu) receive his Ph.D. from Stanfor University in 1969, an has been at Oregon State (Corvallis, OR 97331) since then, except for sabbaticals at Colorao State an Montana State an frequent visits to the University of Montreal. His research interests inclue the stuy of the existence, uniqueness, an approximation of solutions to bounary value problems for ifferential an integral equations arising in mathematical physics. He thoroughly enjoys teaching an has a long-term interest in how calculus is taught to science an engineering stuents. We present a new proof of the ifferentiability of exponential functions. It is base entirely on methos of ifferential calculus. No current or recent calculus text gives or cites a proof of the ifferentiability that epens only on such elementary tools. Our proof makes it possible to give a comprehensive treatment of the erivative properties of exponential an logarithmic functions in that orer in ifferential calculus, builing on the stanar introuction to these topics in precalculus courses. This is the logical orer an has consierable peagogical merit. Most calculus books efer the treatment of exponential an logarithmic functions to integral calculus in orer to prove ifferentiability. A few texts introuce these topics in ifferential calculus uner the heaing of early transcenentals but efer the proof of ifferentiability to integral calculus. Both approaches have serious peagogical faults, which are iscusse later in this paper. Our proof that exponential functions are ifferentiable provies the missing link that legitimizes the early transcenentals presentation. Preliminaries We assume that a r has been efine for a > 0anr rational in a precalculus course an that the familiar rules of exponents are known to hol for rational exponents. It is natural to efine a x for a > 0anx irrational as the limit of a r as r x through the rationals. In this way, a x is efine for all real x. 388 c THE MATHEMATICAL ASSOCIATION OF AMERICA
Basic properties of a x for real x are inherite by limit passages from corresponing properties of a r for r rational. These properties inclue the rules of exponents with real exponents an a x is positive an continuous, a x is increasing if a > 1, a x is ecreasing if a < 1. It is not especially ifficult to justify the efinition of a x for x irrational an to erive the foregoing properties of a x for x real, but there are a lot of small steps. A program along these lines is carrie out by Courant in [2, pp. 69 70]. The general iea of each step is well within the grasp of stuents in typical calculus classes. However, just as properties of a r with r rational are routinely state without proof, it is better to give just an overview of the basic properties of a x with x real, illustrate with graphs, an move on to the question of ifferentiability, which is more central to ifferential calculus. A more complete evelopment, beginning with the erivation of properties of a r with r rational, might be given in an honors class. The properties can be extene to a x with x real with the ai of the ensity of the rationals in the reals an the squeeze laws for limits. The conclusion that a x with a > 1 is increasing also relies on the following proposition which shoul seem evient from graphical consierations: If f is a continuous function on a real interval I an f is increasing on the rational numbers in I, then f is increasing on I. The same proposition will provie a key step in the proof that a x is ifferentiable. Henceforth, we restrict our attention to properties of a x with a > 1. Corresponing properties of a x with 0 < a < 1 follow from a x = (1/a) x. The ifferentiability of a x Consier an exponential function a x with any a > 1. In orer to prove that a x is ifferentiable for all x, the main task is to prove that it is ifferentiable at x = 0. Our proof of this epens only on methos of ifferential calculus. It is motivate by the fact that the graph of a x (see Figure 1) is concave up, even though this fact is not assume apriori. B A T C x O x Figure 1. Graph of a x with B = (x, a x ) an C = ( x, a x ) for x > 0. VOL. 36, NO. 5, NOVEMBER 2005 THE COLLEGE MATHEMATICS JOURNAL 389
In Figure 1, imagine that x 0 with x > 0anx ecreasing. Then B an C slie along the curve towar A. The upwar bening of the curve seems to imply that slope AB ecreases, slope AC increases, an slope AB slope AC 0. It follows that the slopes of AB an AC approach a common limit, which is the slope of the tangent line T in Figure 1 an the erivative of f (x) = a x at x = 0. This geometric argument will be mae rigorous. The curve in Figure 1 is actually the graph of f (x) = 2 x. The following table gives values of the slopes of AB an AC roune off to two ecimal places. It appears that the slopes of AB an AC approach a common limit, which is f (0) = slope T 0.7. x 1 1/2 1/4 1/8 1/16 1/32 slope AB 1.83.76.72.71.70 slope AC.50.59.64.66.67.69 With this preparation, we are reay to prove that f (x) = a x is ifferentiable at x = 0. The foregoing geometric escription of the proof an the numerical evience shoul be informative an persuasive to stuents, even if they o not follow all the etails of the argument. Theorem 1. f (0) >0. Let f (x) = a x with any a > 1. Then f is ifferentiable at x = 0 an Proof. To express our geometric observations in analytic terms, let In Figure 1, x > 0an m(x) = f (x) f (0) x 0 = ax 1. x slope AB = m(x), slope AC = m( x). We shall prove that, as x 0 with x > 0anx ecreasing, m(x) an m( x) approach a common limit, which is f (0). To begin with, m(x) is continuous because a x is continuous. The crux of the proof, an the only tricky part, is to show that m(x) is increasing on (0, ) an (, 0). We give the proof only for (0, ) since the proof for (, 0) is essentially the same. We show first that m is increasing on the rationals in (0, ). Fix rational numbers r an s with 0 < r < s an let a vary with a 1. Define g(a) = m(s) m(r) = as 1 s Then g(a) is continuous for a 1an ar 1. r g (a) = a s 1 a r 1 > 0 fora > 1. 390 c THE MATHEMATICAL ASSOCIATION OF AMERICA
Thus, g(a) increases as a increases an g(a) > g(1) = 0 for a > 1, so m(r) <m(s) for 0 < r < s. Thus, m(x) is continuous on (0, ) an m(x) increases on the rational numbers in (0, ). As note earlier, this implies that m(x) increases on (0, ). The argument for the interval (, 0) is similar. For x > 0, Let x 0 with x ecreasing. Then m( x) = m(x)a x, 0 < m( x) <m(x), 0 < m(x) m( x) = m(x) ( 1 a x). m(x) ecreases, m( x) increases, m(x) m( x) 0. It follows that m(x) an m( x) approach a common limit as x 0, which is f (0). Furthermore, 0 < m( x) < f (0) <m(x) for x > 0, which implies that f (0) >0. We believe that this proof is new. We have been unable to fin any other proof that epens only on methos of ifferential calculus. However, the interplay between convexity an ifferentiability has a long history, an we recommen Chapter 1 of Artin [1] to intereste reaers. Theorem 1 an familiar reasoning give the principal result on the ifferentiability of exponential functions. Theorem 2. Let f (x) = a x with any a > 1. Then f is ifferentiable for all x an f (x) = f (0)a x. It follows that f (x) = f (0) 2 a x > 0, an f (x) = a x is concave up, as anticipate. By routine arguments, a x as x, a x 0 as x. The intermeiate value theorem implies that the range of a x is (0, ). The natural exponential function e x The next step in the evelopment of properties of erivatives of exponential functions is to efine e, the base of the natural exponential function e x, within the context of ifferential calculus. Different authors efine e in various ways. Some of the efinitions involve more avance concepts. We prefer a efinition of e base on an important property of e x, namely that e is the unique number for which = 1. x ex VOL. 36, NO. 5, NOVEMBER 2005 THE COLLEGE MATHEMATICS JOURNAL 391
In view of Theorem 2, an equivalent property is x ex = e x for all x. It is not ifficult to justify the efinition of e an, at the same time, to fin an explicit formula for e. To begin with, consier any base a > 1. Since 2 x is increasing an has range (0, ), there is a unique number c > 0 such that a = 2 c. By Theorem 1, x ax a x 1 2 cx 1 = lim = c lim = cm, x 0 x cx 0 cx where a = 2 c an m = x 2x 2 x 1 = lim. x 0 x Observe that x ax = 1 only for c = 1/m an a = 2 1/m. Therefore, e = 2 1/m. Since m 0.7 by previous calculations, e 2.7. Of course, there are much better approximations for e. Since e is one particular value of a, the function e x has all the properties mentione earlier for general exponential functions a x with a > 1. Thus, e x is increasing an concave up, e x as x, e x 0 as x, an the range of e x is (0, ). With this founation, all relevant applications of exponential functions become available in ifferential calculus. Logarithmic functions Once the basic properties of exponential functions have been establishe, it is easy to introuce logarithmic functions as corresponing inverse functions an to evelop their relevant properties within ifferential calculus. The natural logarithmic function (or natural log) is efine by y = ln x x = e y. The erivative rule for inverse functions implies that y = ln x is ifferentiable an Thus, y ln x = x x = 1 x/y = 1 e = 1 y x. x ln x = 1 x. 392 c THE MATHEMATICAL ASSOCIATION OF AMERICA
The familiar algebraic properties an asymptotic properties of logarithmic functions follow easily from corresponing algebraic rules of exponents an asymptotic properties of exponential functions. In typical textbooks that efer exponential an logarithmic functions to integral calculus, proofs of algebraic properties of ln x are base on the uniqueness of solutions to initial value problems an are less informative for most first-year calculus stuents. Comparisons It is worthwhile to contrast our approach with current practices. Most mainstream calculus texts, such as [5]an[8], efer the entire iscussion of exponential an logarithmic functions to integral calculus, where exponential functions are expresse as inverses of logarithmic functions in orer to establish their ifferentiability. As we wrote earlier, this is the reverse of the natural orer. It has the unfortunate consequence that exponential functions are often efine in two ifferent ways that ultimately have to be reconcile. The upshot is a circuitous argument that blurs the istinction between efinitions an conclusions. Moreover, a substantial block of material about erivatives an rates of change is presente in integral calculus, instea of in its natural place in ifferential calculus. Exponential an logarithmic functions have many important applications, such as motion with resistance, that belong in ifferential calculus. A few books, often calle early transcenentals texts, introuce exponential an logarithmic functions in ifferential calculus. Although this arrangement is an improvement over the stanar approach, the ifferentiability of exponential functions remains a stumbling block. Some of these books, such as [3]an[4], simply isplay a little numerical evience an/or a plausibility argument in support of ifferentiability an then assume ifferentiability thereafter. Others, such as [6] an[7], start out with plausibility arguments in ifferential calculus an then return to the subject in integral calculus where proofs are given. Neither alternative is really satisfactory. It is much better, both logically an peagogically, to settle the question of ifferentiability when the issue arises. Our proof that exponential functions are ifferentiable makes it possible to give a mathematically complete early transcenentals presentation of exponential an logarithmic functions in ifferential calculus. Later on, when methos of integral calculus are applie to exponential an logarithmic functions, progress will not be impee by unfinishe business in ifferential calculus. References 1. E. Artin, The Gamma Function, Holt, Rinehart an Winston, 1964. 2. R. Courant, Differential an Integral Calculus, vol. 1, Interscience, 1959. 3. D. Hughes-Hallett, A. M. Gleason, W. G. McCallum, et al., Calculus, 3r e., Wiley, 2002. 4. R. Larson, R. P. Hostetler, an B. H. Ewars, Calculus, Early Transcenental Functions, 3r e., Houghton Mifflin, 2003. 5. S. Salas, E. Hille, an E. Etgen, Calculus, One an Several Variables, 9th e., Wiley, 2003. 6. J. Stewart, Calculus, Early Transcenentals, 5th e., Brooks/Cole, 1999. 7. M. J. Strauss, G. L. Braley,an K. J. Smith, Calculus, 3r e., Prentice Hall, 2002. 8. D. Varberg, E. J. Purcell, an S. E. Rigon, Calculus, 8th e., Prentice Hall, 2000. VOL. 36, NO. 5, NOVEMBER 2005 THE COLLEGE MATHEMATICS JOURNAL 393