The other day, my son asked me how to differentiate \(f(x) = \pi^x \). I thought about it for a minute. Then another. To my surprise I realized I would not be able to explain it to him -- just yet.
He had recently been studying the chain rule, and of course knew that the derivative of \(f(x) = e^x \) was \(e^x \). But that was not going to be enough to differentiate \(\pi^x \).
The problem was that he would have to know the logarithm function, which he hadn't studied yet. Clearly a break from the video games was called for.
A discussion of inverse functions, and the logarithm in particular would be needed. (He is in the 5th grade, so the relatively small gap in knowledge is understandable.
So there followed the requisite discussion of the logarithm function. And that lead naturally to a discussion of how, back in the day, all students actually used tables of logarithms, and their close relative, the slide rule, for scientific calculations. Calculators had not been invented.
The slide rule is based on logarithms, but is much faster. Unless your name is Hans Bethe, that is. Bethe is said to have memorized the log tables, which allowed him to be incredibly adept at calculation. He was the guy who first explained why stars burn, by working out the nuclear reactions in the stars. He did much more, but that always impressed me. Taking the relatively new field of nuclear physics, in which he was a pioneer, and applying it to something like the sun.
So my son is learning about logarithm tables and the slide rule. Why not just use a calculator? Because logarithms and the slide rule require a knowledge of numbers. A knowledge of exponential functions and their inverses. Knowing about them is pedagogical. Which leads back to being able to calculate the derivative of \(\pi^x \).
Simple things like why, in scientific notation, do we insist the decimal is placed after the first digit (why not the second or third?), become clear. Because when calculating logarithms, it is much more convenient when the decimal is after the first digit. We are primarily interested in the mantissa.
Significant figures become easier to motivate. An electronic calculator, with its 17 or 18 decimal accuracy, can obscure this need to not overestimate the accuracy of our measurements, and hence calculations..
My son also has a keen interest in politics, and the shenanigans of the current election. When he finally is old enough to vote, I believe Heinlein would have approved.