I'm new to DnD and my drow character should (if I got that right) know Elven and Common as by the PHB definition of a drow PC. Also, I'd like to take duergar or undercommon as an additional language because of my character background of trading slaves to the grey dwarves.
Does this character also know drow sign-language, maybe as part of undercommon or elven, or would this classify as its own language?
I've come across the mention of drow sign-language when I was confused about why my PC would not know undercommon. I got it now that drow usually speak an elven dialect with undercommon only as a traders language (and I do understand that I can choose undercommon as language from my background). References where I read of this sign language are amongst others the following from the Forgotten Realms wikia:
Drow Sign Language is a silent hand code used to communicate in the Underdark. Unlike other languages that must be heard to be comprehended, Drow Sign must be seen to gather meaning. It has no alphabet or written form. Humans and other goodly races often confuse the language's motions as spell conjurations. Drow Sign Language is commonly known by drow, who find it particularly useful on patrols and in other instances when silence is expedient. It is a very expressive language, able to convey tone and emotions.
And a previously asked question on this site: Can anyone else learn Drow Sign Language?
It seems to be a thing according to the internet, but I believe it is not mentioned in the PHB. The first link also contains some reference links, which I did not yet took a closer look at.
Also, R.A. Salvatore seems to have said that all drow know sign language. I do not know how much of an authority he is, but according to your comments, it seems like he is the authority (?)
Best Answer
Drow sign language isn't an officially included language in D&D 5e rules, at least not as far as I can find.
The languages a PC knows are determined by their race and, save for a special few, are mostly listed on page 123 of the PHB. You'll notice that's there's no mention of a drow sign language. It's not even mentioned in the Sword Coast Adventurer's Guide (a 5e sourcebook for campaigns set in the Forgotten Realms).
If you also look at the references for the Wikia page you've linked you'll notice that it's mostly referencing novels by R.A Salvatore set in the Forgotten Realms and previous editions of D&D. In general, a novel does not equal a rules books for a TTRPG. Even the other question from this site that you've linked is asking about 3.5e, not 5e.
In terms of how much influence a novel does have on rules created for a TTRPG, well, that will be determined by your DM and the world they set their campaign in.
However, this doesn't mean your drow can't know drow sign language. It just means that, by default, it doesn't know it because it's not a part of the rules.
A DM is free to create their campaign setting, their world, their universe, their multiverse, however they please. If you want your drow the know sign language, talk with the DM; see if it's something they think would make sense in their world and if they can incorporate it.
The DMG even has a section on languages, from page 20-21: