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So, I recently decided to major in mathematics, and I take calculus next semester, but I bought a calculus textbook and have been learning a little bit of calculus. I was wondering if I understand this correctly, but the derivative is not the instantaneous rate of change but rather the best constant approximation for the rate of change. Instantaneous means "no lapse in time". To me that makes the definition of a derivative nonsensical because you can't measure a rate of change when there is no lapse in time.

J. W. Tanner
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Tom
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    When driving a car, do you disagree that you can look at the speedometer and look to see how fast you are going at that moment? – JMoravitz Jan 04 '23 at 03:10
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    The derivative is the limit of the rate of change as the lapse in time approaches zero – J. W. Tanner Jan 04 '23 at 03:11
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    derivative is the best aproximation by a linear funciont to your desired function – L F Jan 04 '23 at 03:12
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    Perhaps your worry is that if we look at "no lapse in time" we end up with the average rate of change formula giving a "zero divided by zero" result... but (and here's the kicker)... so long as we are careful and we instead talk about the limit as the lapse in time merely approaches zero... then we aren't talking about zero divided by zero at all! Rather, we are talking about a small number divided by another small number... but where these are proportional to each other in such a way that we can safely perform the algebra in a meaningful manner that gives meaningful answers! – JMoravitz Jan 04 '23 at 03:15
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    While mathematics is a fine subject to study, I will suggest you do not commit to anything until you have a couple of courses taught rigorously (i.e. with a focus on proofs). – user317176 Jan 04 '23 at 03:17
  • @DougM I appreciate the advice, but I have thought long and hard about this and this is definitely what I want to study. Personally, I absolutely love proofs and my concentration is in pure mathematics and mathematics teaching (I plan on becoming a teacher). – Tom Jan 04 '23 at 03:23
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    Perhaps this video by 3Blue1Brown will prove enlightening. – PrincessEev Jan 04 '23 at 03:29
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    See my answer here, where I show that the tangent is the best linear approximation to a function at a point: https://math.stackexchange.com/questions/1784262/how-is-the-derivative-truly-literally-the-best-linear-approximation-near-a-po/1784303#1784303 – marty cohen Jan 04 '23 at 04:09
  • "instantaneous" is superfluous. – copper.hat Jan 04 '23 at 04:24
  • You can think of it as an infinitesimal lapse in time. And, yes, infinitesimals have been rigorously added to mathematics since the 1960s. – johnnyb Jan 04 '23 at 19:42

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