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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.2011-03-13
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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.2011-03-13
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    One difficulty is that most calculus books have "proved" certain facts about derivatives and integrals, but not enough facts about analysis to establish something like continuity of a limit of a sequence of functions. In particular, they would have to establish some kind of uniform convergence of the sequence (because pointwise convergence doesn't preserve continuity). That sort of thing would be a diversion from other material, and pedagogically difficult. The advantage of the $1/x$ method is that it is relatively self contained given the other material in a calculus course.2011-03-13
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    On the other hand, the definition via a power series, once the basic facts about power series are known, is again easier to deal with than a definition in terms of a convergent sequence of other functions. For example, computing the derivative becomes just an application of a general theorem.2011-03-13
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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.2011-03-13
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    Where's the spoiler warning for that remark about Rosebud? Not everyone's seen that film, you know... ;)2011-04-28
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    @Peter: Yes, and to quote Lisa Simpson: "And the chick in 'The Crying Game' is a guy!"2011-04-28
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    A (relatively) quick way to see that $x\mapsto k^x$ extends to the irrationals is that it is convex or, equivalently, $x\mapsto k^{x+h}-k^x=k^x(k^h-1)$ is increasing for any $h > 0$. This also guarantees that it has right and left derivatives everywhere and, then, that it is differentiable.2011-05-26

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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.