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The story in nearly every introductory Calculus book is well known by everybody: you don't have the "right" to raise a number to an irrational power, so forget exponents for now and let's take a look at $y=x^{-1}$. How odd, the innocent formula for a power function's antiderivative breaks down but gee, it must have an antiderivative, it's smooth! Let's examine its properties...

...and in the end, Rosebud was his sled, no, wait, the mysterious antiderivative turns out to have an inverse that corresponds exactly to the elementary school concept of exponents, only it works for irrational exponents too! The hero wins! The End.

But what if we start from the opposite end? Start with the innocent, only-defined-for-rationals (so far) exponential function $y=k^x$, $k>0$, and if $x_0$ is irrational, prove that, as $x$ (while staying rational) approaches $x_0$, $k^x$ approaches some specific real number. Define that such number is $k^{x_0}$.

And from there, prove that our New! Improved! $k^x$ is continuous, has a derivative that's also an exponential, that there is some $k=e$ for which the exponential is its own derivative, that the inverse of $e^x$ has $x^{-1}$ as its derivative etc etc...

Do you know of any Calculus text that takes that approach?

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Comment: I'd imagine almost any real analysis book would take the second approach, e.g. Rudin. –  charles.y.zheng Mar 13 '11 at 1:36
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Like charles.y.zheng says, Rudin's Principles of mathematical analysis has this approach in an exercise set. On the other hand, the exponential function is sometimes simply defined by its power series, e.g. in Rudin's Real and complex analysis and typically in complex analysis texts. –  Jonas Meyer Mar 13 '11 at 1:54
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Almost all calculus textbooks that have "early transcendentals" begin by "defining" (pseudo-defining, really) the exponentials, and defining the logarithms as their inverses, not the other way around. Because they treat derivatives (including derivatives of logarithms and exponentials) well before they introduce the notion of anti-derivatives. –  Arturo Magidin Mar 13 '11 at 3:21
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Where's the spoiler warning for that remark about Rosebud? Not everyone's seen that film, you know... ;) –  Peter Taylor Apr 28 '11 at 12:16
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@Peter: Yes, and to quote Lisa Simpson: "And the chick in 'The Crying Game' is a guy!" –  Arturo Magidin Apr 28 '11 at 13:38

3 Answers 3

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I like the approach Lang takes in Undergraduate Analysis. He defines the exponential as a function that satisfies the following differential equation subject to specified initial conditions:

$$ f^{\prime} = f, \;f(0)=1 $$

Using these assumptions he shows that if $f$ exists then it is unique. Later in the text he proves existence with power series. He gives an analagous treatment for $sin(x)$ and $cos(x)$. Fitzpatrick's Advanced Calculus takes a similar approach

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Maybe you need the G.M. Fikhtengolts's book A course in differential and integral calculus (Фихтенгольц Г.М.: Курс дифференциального и интегрального исчисления)? The vol.1 give a construction from the irrational power.

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I have seen other ways to do it. I know at least one analysis course at my university (for bio-engineers) defines the exponential function through its Taylor series and works its way from there.

The other approach I've seen extends from rational to real exponents by using Cauchy sequences of rational numbers.

I must say neither of them are basic calculus texts, but they are still basic analysis texts.

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