Hmm... I see.. you think the article in Wikipedia is lacking precision? He went on this line of thought:

"The differential of a single-variable function ##f(x)## is a two-variable function ##df(x,\Delta x)## defined as

##df(x,\Delta x) = f'(x)*\Delta x##"

Then he says "one or two arguments may be suppressed", that is, ##df(x,\Delta x)=df(x)=df##. I understood he's saying the 3 forms are the same thing, ergo defined the same way.

Then he goes on saying "since ##dx(x,\Delta x)=\Delta x##" it is "conventional" to write ##dx=\Delta x##. Now, that "conventional" is confusing to me. I first interpreted what he said as this... you take a variable x, then build a function ##X(x) = x##, so that now we can define a differential of that function as ##dX(x,\Delta x)=X'*\Delta x=\Delta x##, and then the "convention" was that whenever one refers to "##dx##" he really means "##dX(x,\Delta x)=\Delta x##" - under a limit, of course.

But I get that this is not right, so I think I just don't get the nature of ##dx##. How would you describe the nature of ##dx##? That's a variable, a function, or an operator?

Or, better yet, using the language the Fresh_42 taught me... what is the domain of ##dx##?

"The differential of a single-variable function ##f(x)## is a two-variable function ##df(x,\Delta x)## defined as

##df(x,\Delta x) = f'(x)*\Delta x##"

Then he says "one or two arguments may be suppressed", that is, ##df(x,\Delta x)=df(x)=df##. I understood he's saying the 3 forms are the same thing, ergo defined the same way.

Then he goes on saying "since ##dx(x,\Delta x)=\Delta x##" it is "conventional" to write ##dx=\Delta x##. Now, that "conventional" is confusing to me. I first interpreted what he said as this... you take a variable x, then build a function ##X(x) = x##, so that now we can define a differential of that function as ##dX(x,\Delta x)=X'*\Delta x=\Delta x##, and then the "convention" was that whenever one refers to "##dx##" he really means "##dX(x,\Delta x)=\Delta x##" - under a limit, of course.

But I get that this is not right, so I think I just don't get the nature of ##dx##. How would you describe the nature of ##dx##? That's a variable, a function, or an operator?

Or, better yet, using the language the Fresh_42 taught me... what is the domain of ##dx##?

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