It’s pretty common to hear that dy/dx isn’t a fraction, but if that’s the case then why do we treat as such in a differential equation or an integral? For example, if dy/dx = f(x), then how can we just write it as dy= f(x)dx as though it’s a fraction?
In: Mathematics
If you asked a pure mathematician they’d probably tell you never to write:
> dy = f(x) dx
A “dx” thing should never be on its own. You should write:
> ∫ dy = ∫ f(x) dx
The “∫” goes with the “dx”; they are two parts of the same thing. Both “∫[thing] dx” and “d[thing]/dx” are *operators.* They are things we do to functions. They mean “take the function and integrate it with respect to x” and “take the function and differentiate it with respect to x.”
*However*, there are some situations where they do behave similarly enough to fractions that we can kind of treat them like that. If we are careful.
dy/dx can be defined as a limit of a fraction:
> dy/dx = limit as Δx -> 0 of Δy / Δx
where Δy and Δx are separate things. So in the right circumstances, depending on what that limit is doing, you can treat the dy/dx part like a fraction. But you might upset some pure mathematicians.
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