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In 3blue1brown's video of the paradoxes of the derivative, I'm confused about his overall perspective on the derivative. In the beginning he says that "instantaneous rate of change" is nonsense and he also gives an example "If I show you a picture of a car, a snapshot in an instant, and ask you how fast it’s going, you’d have no way of telling me.". But later he mentions that the derivative is equal to the slope of a line tangent to the graph at a single point or in other words, the derivative is the rate of change at a point. Isn't he contradicting himself?

Pls correct me if I'm wrong but my understanding is that he is trying to say that the derivative is the rate of change at an immediate point but computing the derivative requires an interval that is constantly getting smaller and smaller. The rate of change that the interval constantly approaches is the derivative.

mathisconfusingaf
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  • I think what he is saying is that if you actually wanted to measure the speed of the car in real life, you would have to record the car's motion over some really small (but non-zero) snapshot of time. So the derivative is an on paper theoretical concept, but still a very important concept. The derivative tells what you'd get for the speed of the car in an idealized world where you could measure the displacement of the car over an infinitesimally small time interval. – Joseph Robert Jepson Mar 17 '23 at 01:24
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    I have encountered that perspective before but I disagree. Finding the limit gives the snapshot of motion. The velocity vector represents the speed you were going and the direction you were going when the clock stopped. The smaller intervals we use to compute it don't represent its value; the limit does. – John Douma Mar 17 '23 at 01:50

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