Where can/should I, as a high schooler, publish a paper regarding the derivation of a formula?

The big question here is whether what you've done is new to the world - whether it's a genuine new advance that nobody in the world has done before, or whether it's something known to the world that you, and perhaps your teacher, have not seen before. Without meaning any disrespect to you and your friend, the second option here is more likely - but the first is of course possible.

If it's genuinely new, you can look to publish in a scientific journal. If it's not new (but perhaps still unusually advanced for a high school student), you should perhaps look for other types of publication, which could still be good for your resume. The difficult question is how to tell which category your work falls into.

My suggestion is to speak to your high school teacher about this. They have seen a lot of different students' work. If they think that it may be a novel development, then either write, or get your teacher to write, to a professor in a relevant subject at a local university. See if they will look it over and advise you.

Check out the American Journal of Undergraduate Research. You need a faculty mentor. Make contact with your state university.

Do plenty of online searches to see if your work is already published. This will take some time. Use various keywords.

Even if your formula or equation is not new, if the derivation is new, you can still publish. This is especially true if your derivation yields new insights into the result, or if the derivation serves as a model to solve harder problems. So take a look at the derivation to see if it yields any insights to harder problems.

If you have something publishable, don't give up. Be persistent. This may not happen for a while, but keep at it.

Do you both have any contacts in the engineering field? They may be able to situate your issue and have contacts that might be able to help.

It is a big ask for high school students I know but try to spend the time to do the research if you can? Paywalls to journals are a big problem. What else has been done around your issue and question. What other available equations can be used? What do people do now to sort out this problem? This would give you a clue to who and what sort of people/researchers are tackling the problem and would be interested in your equation if it is indeed unique. Even if it is not unique, the contacts may also become a useful resource for your future.

Physics SE have not addressed your question and I suspect that your equation addresses a pragmatic issue rather than a theoretical problem so maybe the Engineering StackExchange might be a better fit. However, spend the time going through the various tags and branches in the SE rather than just jumping from one StackExchange to another. Write a couple of paragraphs about the current approaches available, quote other relevant questions or sites that explain the issue in better detail and explain how your equation is unique. This skill of "situating" the problem and being to respectful to others trying to solve the same problem while discussing the background in an engaging way is an important and underappreciated part of "doing research".