7

« **on:** October 15, 2018, 06:06:34 pm »
Here's something I've been thinking about again recently

So, a while ago I tried to prove that the derivative of sin is cos using the limit calculation, that's lim h->0 [sin(x + h) - sin(x)] / h. After finishing it, I realized that I used the rule of L'hopital, so I used the fact that sin' = cos in order to prove that sin' = cos. And ofc logically speaking, sin' = cos => sin' = cos is a tautology, so it doesn't prove anything. But then I realized that it's actually still pretty strong evidence – certainly it would be strong rational evidence if you didn't know what the derivative of sin was – because if you postulate an incorrect derivative, the same calculation will most likely get you a contradiction. For example, if you postulate that sin(x)' cos(x)' = x, then what you prove is that sin'(x) = cos'(x) = x => sin(x) = 0, which is a contradiction, so it does give you a valid proof that sin(x)' or cos(x)' does not equal x.

This made me think that perhaps the correct result is the *only* result that would not yield a contradiction in this way. If that were true and you could somehow prove it to be true, then the tautological proof of sin' = cos would actually become a legit proof. It turns out that's not true, though, because I found two counterexamples: sin' = cos' = 0 and sin' = sin, cos' = cos both return tautologies rather than contradictions. this keeps being true if you also plug them into the limit calculation for cos(x). But I still suspect that the class of contradictions is quite small. maybe that's wrong.

It also made me think about whether this is a formally stateable question. You may not be able to ask "is there another function f such that if sin(x)' = f(x), you get a stable result doing the limit calculation with l'hopital", because if sin(x)' = f(x) for f(x) ≠ cos(x), you would have a contradiction and then everything follows, so it probably *would* be possible to do the calculation and conclude that sin(x)' = x => sin(x)' = x. This is the general problem of reasoning about logical uncertainty. But at the same time, it is a question that's pretty easy to understand informally. Maybe you could formulate it if you restricted the operations that are allowed, but that sounds weird.