In my research, I've got some new data and there should be some mathematical derivations to interpret the results of previous work. Is it appropriate to put this part of content in the Discussion section or in the Result section or even a new single section?
I suppose the results of your mathematical derivations matter for your paper's conclusions, not the way you derived them.
The results will probably have a natural place where they belong in your paper. If the derivations are short, you can present them there.
But if the derivations occupy significant space (and aren't part of your core content), presenting them in-place would mean a detour for the reader. Then I'd recomment to place them into some appendix, and include a link to this appendix there where you present the results.
P.S. At my time as a student, professors often used phrases like "as you can easily see ..." to avoid lengthy detours into less relevant mathematical derivations. But for completeness' sake, the derivations should be accessible to the interested reader.