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human
Why does Calculus use dx to represent the change in x when other areas of science, such as physics, use delta-x?

I'm taking a Calculus class this year along with a physics class and dx and delta-x seem to represent the same thing. Why are there two different symbols used (d vs. delta)? Is there even a reason?
gpt
They are two different things.

If you have a function, f(x), and two values of x, say x*_1_* and x*_2_*, then ∆f is equal to f(x*_2_*)-f(x*_1_*). In particular,  ∆f/∆x is equal to (f(x*_2_*)-f(x*_1_*))/(x*_2_*-x*_1_*). Mathematically, since x*_1_* and x*_2_* are different, then they are "far" apart. No matter how close they are, there is still space in between them where lots of stuff can happen.

df/dx is the *limit* of ∆f/∆x as both x*_1_* and x*_2_* approach the same point. This is fundamentally different from ∆f/∆x, because limits are involved. In particular, we lose the problem of there being space between the points, at the cost of not knowing what is happening at any point different from x.

However, if the limit df/dx exists, then by the definition of limits, if we're okay with some fixed error, then there will always be some x*_1_* and x*_2_* so that ∆f/∆x approximates df/dx within that error. That is ∆f/∆x ≈ df/dx, where the difference is within some acceptable error. This follows directly from  the epsilon-delta definition of limits (which is probably the most important thing to take away from a calculus course).

As for things like dx and df, you don't really need them. Their actual definition is very abstract and pretty much anytime you see them in physics or engineering, they are used wrong and the expression could be stated differently without them. Anytime you see or use "dx" in a physics or engineering course, take it with a grain of salt because it's probably not representative of what dx should be. For uses in these classes, you can take dx to mean the ∆x so that x*_1_* and x*_2_* are close enough so that ∆f/∆x ≈ df/dx within an acceptable error. This isn't what dx actually is, at all, but it's how it is used in these courses.

**EDIT**: It should be emphasized that df/dx is *not* "df divided by dx", it is a limit of ∆f/∆x and this limit cannot be broken up into a fraction of limits. With the actual definitions of df and dx, you cannot get df/dx by dividing df by dx (division doesn't even make sense for these objects, it's more subtle than that). df/dx is just the notation we use for derivatives, we could always just use f'(x), never even mention dfs or dxs ever in a Calculus 1 or 2 course and we wouldn't miss a thing.