The paradox of the derivative | Chapter 2, Essence of calculus
Derivatives center on the idea of change in an instant, but change happens across time while an instant consists of just one moment. How does that work?
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The goal here is simple: Explain what a derivative is. Thing is, though, there’s some subtlely to this topic, and some potential for paradoxes if you’re not careful, so the secondary goal is that you have some appreciation for what those paradoxes are and how to avoid them. You see, it’s common for people to say that the derivative measures “instantaneous rate of change”, but if you think about it, that phrase is actually an oxymoron: Change is something that happens between separate points in time, and when you blind yourself to all but a single instant, there is no more room for change. You’ll see what I mean as we get into it, and when you appreciate that a phrase like “instantaneous rate of change” is nonsensical, it makes you appreciate how clever the fathers of calculus were...
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The goal here is simple: Explain what a derivative is. Thing is, though, there’s some subtlely to this topic, and some potential for paradoxes if you’re not careful, so the secondary goal is that you have some appreciation for what those paradoxes are and how to avoid them. You see, it’s common for people to say that the derivative measures “instantaneous rate of change”, but if you think about it, that phrase is actually an oxymoron: Change is something that happens between separate points in time, and when you blind yourself to all but a single instant, there is no more room for change. You’ll see what I mean as we get into it, and when you appreciate that a phrase like “instantaneous rate of change” is nonsensical, it makes you appreciate how clever the fathers of calculus were...
