General definition of a derivative

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SUMMARY

The general definition of a derivative is expressed as f'(x) = lim(Δx → 0) (Δy/Δx). This definition emphasizes the limit as Δx approaches zero, rather than Δy, due to the function being represented as y = y(x). The discussion also highlights the potential confusion between the notations f(x) and y(x), clarifying that while f(x) represents a function, y(x) is merely an alternative representation and does not imply a different function. Misinterpretations of these notations are common among students and professionals alike.

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cscott
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I was told that the general definition of a derivative is

f'(x) = \lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}
(supposed to be delta y over delta x, but I can't make the latex work :mad:)

but why can't it work when \Delta y \rightarrow 0?
 
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Because the function is y=y(x),so it's natural to consider the limit on the "x" (variable's) axis.


Daniel.
 
Oh, alright.

Another thing, what is f(x) = y or y(x) = y in normal notation? I thought f(x) replaced y, but the fuction y = y doesn't make sense, does it? I mixed up :frown:
 
Last edited:
Abuse of notation,i don't know how much mathematicians do it,but physicists adore it.

Daniel.
 

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