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While reading through physics history book, i en counted an attempt to show the very basics of Leibniz notation; the following is shown:

If

s = (1/2)gt^2, then

s + ds = (1/2)g(t + dt)^2

s + ds = (1/2)gt^2 + gtdt + (1/2)gdt^2, then because of the first line

ds = gtdt + (1/2)gdt^2

It then goes on to say (1/2)gdt^2 "can be ignored because it is so small". Obviously this is not really correct, and i was looking for a way to explain using limits. I came up with something, but it seems incorrect in some way?

If

ds = gtdt + (1/2)gdt^2, then

(ds/dt) = gt + (1/2)gdt . Let (ds/dt) = f, then

lim f {dt -> 0} = gt

However, the confusion is that this does not seem to imply

(ds/dt) = gt as is normally written; it seems to imply

lim f_{dt -> 0} = lim (ds/dt)_{dt -> 0} = gt

Can somebody please explain how Leibniz would have came to (ds/dt) = gt without "ignoring (1/2)gdt^2" ? Was the notion of a limit used back then?

Thanks in advance,

Brendan

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# Leibniz notation

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