If I rederived a result (oblivious that it already existed), should I cite the original paper?
Citation has multiple purposes, as described by several answers on this site (e.g. 1, 2). Broadly speaking, those purposes mostly fall into three classes: giving a trustworthy source, giving credit, or suggesting further reading. Any individual citation might serve several of these purposes, or only one of them. So when you cite a paper, that doesn't necessarily make it seem like you took the result from there; you might simply be giving credit to the first person(s) to publish the result, or referencing an external source to back up what you're saying.

What this means for your case is that you should definitely cite the other paper. The exact wording you use to do so depends on several factors, including whether you think an explicit statement of the result, and/or your derivation of it, are still worth including in your paper - for example, if you think your derivation is more clear, or relaxes some assumptions of the original, or so on. Here are some possibilities along the lines of what I've used or read:


"This result was originally derived in reference [1]." (possibly in a footnote)
"For a more detailed derivation, see reference [1]."
"Here I present an alternate derivation of the result in reference [1]."
"(statement of result) [1]. In brief, this can be derived as follows: ..."
"This uses the result that (brief statement of result) [1]." (probably in a footnote)


and so on. Or you can just cite it without comment, and unless the conventions of your field are otherwise, people won't necessarily assume that you took the result from the other paper. (Nor will they even care, in most cases.)