Defining the derivative without limits?

Question

These days, the standard way to present differential calculus is by introducing the Cauchy-Weierstrass definition of the limit. One then defines the derivative as a limit, proves results like the Leibniz and chain rules, and uses this machinery to differentiate some simple functions such as polynomials. The purpose of my question is to see what creative alternatives people can describe to this approach. The nature of the question is that there is not going to be a single best answer. I have several methods that I've collected which I'll put in as answers to my own question. It's not reasonable to expect answers to include an entire introductory textbook treatment of differentiation, nor would anyone want to read answers that were that lengthy. A sketch is fine. Lack of rigor is fine. Well known notation and terminology can be assumed. It would be nice to develop things to the point where one can differentiate a polynomial, since that would help to illustrate how your method works and demonstrate that it's usable. For this purpose, it suffices to prove that if $n>0$ is an integer, the derivative of $x^n$ equals $0$ at $0$ and equals $n$ at $1$; the result at other nonzero values of $x$ follows by scaling. Doing this for $n=2$ is fine if the generalization to $n>2$ is obvious.

Answer 1

Suppose ƒ(a) = b and L(x) = b + m(x – a).

Suppose for all x in some open neighborhood of a we have

L(x) < f(x) if x < a and L(x) > f(x) if x > a.

Then m is not too small to be ƒ '(a).

In a similar way, we define what it means to say m is not too big to be ƒ '(a).

If just one number is not too small and not too big, then that number is ƒ '(a).

Answer 2

Nowadays, your best bet maybe do it strictly numerically.

Have them calculate the slope of the secant line on an interval, say h=0.1 wide.

Then compare that to the slope of the tangent line at the midpoint.

Next repeat the same calculations for h = 0.05.

Then again for h=0.01.

Then again for h=0.005.

Keep shrinking h -->0 until the convergence becomes clear....