I have had a number of mathematical epiphanies lately in calculus. A few years back, when I was hacking my way through calculus at the U of M, I never imagined that the derivative of a parametrically defined function would ever make sense to me. I also never found implicit differentiation to be all that exciting. I figured out how to move the numbers and get the answers, but there was no beauty to it. I guess both of those topics seemed like untidy little loose ends that were clumsily dealt with by the big-haired math guys from days of yore. Boy was I wrong. I now feel like I can understand the necessities and the mechanisms for each of these concepts, and I really can appreciate the utter inspiration.
Here's what had been missing: In both cases, we are able to find a derivative dy/dx in spite of the fact that there is not the traditional relationship between independent and dependent variables. In the case of parametric functions, x and y are both dependent variables defined in terms of t, the parameter. In the case of relationships which are implicitly defined, there is not a proper arrangement between independent and dependent variables. In both cases, the graphs of the functions, the patterns of ordered pairs that satisfy the equations, exist on the x-y plane and therefore have slopes which are defined as the change in y compared to the change in x, or dy/dx. The ways in which this quantity is obtained are remarkably clever.
Anyway, most of the people who read this will be the more grateful to be finished with math forever. The rest of them will wonder how I never figured this out until now.
Today for two of my calculus sections, I delivered (with relish) the third installment of the power rule for differentiation. We are now justified in using the rule when the exponent is rational. For my efforts, I got a bunch of unimpressed stares and one "that's the same result we got the first time". Of course it's the same result, isn't that great? But now we have shown that it works for rational numbers. The first time, we were only able to algebraically show that the rule worked whenever the exponent was a positive integer, and our proof made no sense for other powers. After that, we used the quotient rule along with the positive power result to show that negative integer powers give the same rule. Now we have used implicit differentiation and both previous results to show that the exponent can be any rational number! Next class we'll prove it for the reals!
Uhhh... Mr. Burchell, why didn't you just tell us the real number thing at the beginning?
From my point of view, we are using our elementary tools to construct bigger and better tools, creating the amazing structure of Calculus from simplicity itself. Some of the students wonder why we don't just start with the awesomest tools. Because if we did, it would not be awesome.