Explaining dy, dx, ds, dt & More in Calculus & Physics

[Hyperreality]

From what I've learned in the first year of calculus, a derivative is always written in the form of dy/dx and it must be written in this form, for dy and dx to have any mathematicl meaning.

But in many physics textbook I've been self studying from, dy, dx, ds, dt... of such have been used frequently. My guess is that it has something to do with the "geometrical aspect" of the variable, where can I find a definite explanation for these notations?

For instance the, the intensity in the range of \lambda + d\lambda,

dI = R(\lambda)d\lambda

Why can't they just write as

dI/d\lambda = R(\lambda)?

 

[courtrigrad]

They are using the concepts of differentials

 

[HallsofIvy]

The derivative "df/dx" (also denoted f ') is defined as a LIMIT of a fraction ((f(x+h)- f(x))/h) so it is not a fraction itself: dy and dx separately make no sense.

HOWEVER, since you can always go back "before" the limit in proofs and use the fraction properties of (f(x+h)- f(x))/h, df/dx "acts like" a limit. To make use of that,
we can define the "differential": df is defined as f '(x)dx and dx is, essentially, left undefined (think of it simply as a notation). One result of that is you should never have a 'dy' without a corresponding 'dx'. Some textbooks immediately point out that you can approximate dx by a small number (Δ x) but, unfortunately, many students interpret that to mean dx is a small number (as we've seen several times on this forum).