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 (?)