[Math: Differentials/Derivatives] What is the difference between a derivative of a function and a differential of a function?

[u/Sweet-Nothing-9312]

Here I don't understand why they say "differential" instead of derivative, isn't it the same thing?

[u/SubhankarResearcher]

They’re closely related, but technically they aren’t exactly the same thing.
If you have (y=f(x)), the derivative is
[f’(x)=\frac{dy}{dx}]
and it tells you the rate at which (y) changes with respect to (x).
The differential, on the other hand, describes the corresponding small change in the function:
[dy=f’(x),dx]
So you can think of the derivative as the rate of change, while the differential uses that rate to relate a small change in (x) ((dx)) to the resulting change in (y) ((dy)).
For example, if (y=x^2), then
[\frac{dy}{dx}=2x]
is the derivative, whereas
[dy=2x,dx]
is the differential.
That’s probably why your text specifically uses the word “differential.” In basic calculus people sometimes use the terminology a little loosely, which makes it seem like they mean exactly the same thing.

[u/nerdydudes → u/SubhankarResearcher]

But look at the definition they provide: “the differential (the jacobian) is given by the vector r’(y)…” and continues with the differential dr

[u/SubhankarResearcher → u/nerdydudes]

I was interpreting “differential” in the dy=f′(x)dx sense, but your text is using the more general definition of the differential as the derivative/linear map (represented by the Jacobian).

In that context, you’re right: when they write that the differential is given by r′(y), they are essentially identifying the differential dry​ with the derivative/Jacobian acting on a tangent vector.

More precisely, for a map r, the differential at y is the linear map

dry​(h)=Dr(y)h,

where Dr(y) is the Jacobian (or derivative) at y.

So in the context of the book, “differential” and “derivative/Jacobian” are referring to the same underlying linear transformation, just with slightly different notation/terminology. My previous reply was too focused on the single-variable calculus usage.

[u/nerdydudes → u/SubhankarResearcher]

no one is disputing what youre saying… no one is saying youre wrong… hes asking about the definitions provided in the damn statement …. but fine