Don Knuth says we should teach calculus without limits.
I would define the derivative by first defining what might be called a “strong derivative”: The function has a strong derivative at point if
I think this underestimates the difficulty for novices of implicit definitions. We’re quite used to them: ”f’(x) is the number such that bla bla, if such a number exists, and, by the way, if such a number exists it is unique.” Students are used to definitions that say, simply, “f’(x) is bla.”
Now I will admit that the usual limit definition has hidden within it an implicit definition of the above kind; but I think the notion of limit is “physical” enough that the implicitness is hidden from the eyes of the student who is willing to understand the derivative as “the number the slope of the chord approaches as the chord gets shorter and shorter.”
Another view — for many if not most calculus students, the definition of the derivative is a collection of formal rules, one for each type of “primitive” function (polynomials, trigonometric, exponential) together with a collection of combination rules (product rule, chain rule) which allow differentiation of arbitrary closed-form functions. For these students, there is perhaps little difference between setting up “h goes to 0″ foundations and “O(eps)” foundations. Either set of foundations will be quickly forgotten.
The fact that implicit definitions are hard doesn’t mean we shouldn’t teach them to first-year college students, of course! Knuth is right that the Landau notation is more likely to mesh with other things a calculus student is likely to encounter, simultaneously with calculus or in later years. But Knuth seems to say that big-O calculus would be self-evidently easier and more intuitive, and I don’t think that’s evident at all.
Maybe we could get students over the hump of implicit definitions by means of Frost:
Home is the place where, when you have to go there,
They have to take you in.
(Though it’s not clear the implied uniqueness in this definition is fully justified.)
If I were going to change one thing about the standard calculus sequence, by the way, it would be to do much more Fourier series and much less Taylor series.