Calculus I · The Function That Is Its Own Slope

The honest part: what we borrowed, and when we pay

This is the section most textbooks leave out, and I think leaving it out is a quiet form of lying. So, plainly: where in this chapter did we lean on something we had not proven?

1. We assumed the limit $M(a) = \lim_{h\to 0}\frac{a^h - 1}{h}$ actually exists — that the curve $a^x$ has a definite slope at the axis. The numbers behaved, and the picture is convincing, but “the numbers behaved” is not a proof.
2. We assumed $M(a)$ slides smoothly from below $1$ (at $a=2$) to above $1$ (at $a=3$), so that some base hits slope exactly $1$. Again: true, pictured honestly, not yet proven.
3. We took on faith that this $e$ — “the slope-$1$ base” — is the same number as Chapter 1’s $\left(1+\frac1n\right)^n$.

None of these is wrong. All three are repairable. But the clean, no-loans way to build $e$ runs in the opposite direction from how we did it: you first define $\ln x$ as the area under the curve $1/t$ from $1$ to $x$ — an integral — prove from that definition that it turns multiplication into addition and has slope $1/x$, and then define $e^x$ as its inverse and $e$ as the input where $\ln$ equals $1$. Built that way, every fact above becomes a theorem with no gaps. We cannot do it yet because we have not built integrals. We build them in Chapters 9 and 10, and the very last thing Chapter 10 does is come back here and discharge all three loans at once.

Why borrow at all, then? Because waiting would have meant not touching exponentials — the engine of half the science you care about — until the end of the book. Mathematicians borrow like this constantly: you convince yourself a tool is sound, you use it to get real work done, and you return to nail down the foundations once you have the right machinery. Watching where the loan sits, and never forgetting to repay it, is itself one of the skills this series is trying to teach you.

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