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I understand how to solve these problems. I know that the derivative means the slope and stuff. Just what I truly want to know is why. If you were living a few hundred years ago, and you graphed velocity = t^2 and randomly decided to plot the slope of the velocity of an object, how would you come to realize that the derivative, 2t, is telling you the acceleration?

StackethMatheth
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    This is largely how Newton and Leibniz and their promoters explained their calculus results a few hundred years ago. – Henry Oct 01 '24 at 19:48
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    People often do things with a purpose rather than randomly. First you have to think about what a graph of velocity as a function of time is telling you. Some hints: what does it mean that the velocity is different at different times? Is the velocity changing more at some times than others? If you wanted to quantify that, how would you do it? – Will Orrick Oct 01 '24 at 19:50

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That comes from the understanding of what velocity and acceleration are. Their bare definitions I mean. The velocity is the (instantaneous) rate of change of the position, and the acceleration is the (instantaneous) rate of change of the velocity.

nvimtyper
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