Best answer:
Suppose you're on a hill and you want to calculate the slope of it (i.e. how much it rises for a given horizontal distance).
Well, you can take two points on the hill and divide rise with run.
But what if the hill is super bumpy and uneven? The method gives the average slope between two chosen points, but what...
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Best answer: Suppose you're on a hill and you want to calculate the slope of it (i.e. how much it rises for a given horizontal distance).
Well, you can take two points on the hill and divide rise with run.
But what if the hill is super bumpy and uneven? The method gives the average slope between two chosen points, but what if you want the slope at the *exact* point you're standing at? Like if you were to put a ball at your feet, would it roll off or not?
Obviously you want to reduce the "averageness" in your calculation which is done by bringing the two points closer together so it more closely approximates the exact slope where you're standing.
By bringing the points closer and closer your answer gets more accurate, and when you bring them SO close that they become the SAME point, then the slope is exact. And that is a derivative.
For this much closeness we need some sort of limit in our definition of derivative because we are bringing Δx to 0 and a limit is the only thing that allows you to divide with 0 while still making sense.
So there is obviously a difference between an *approximation* where Δx is small but not 0, and the exact answer where Δx ---> 0. In that context Δx turns to dx so you can distinguish between the two.
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2 days ago