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In Harry Potter and The Chamber of Secrets, Draco calls Hermione 'a filthy little mudblood'. She never let slip she was muggleborn in the Philosopher's Stone book or film, so how does he know Hermione was muggleborn?
My theory is that Lucius Malfoy (who was a governor at the time) would have told Draco who all the muggleborns are.
I'm not sure why you think Hermione "never let slip" that she was Muggle-born. She talks about it freely the very first time we meet her:
"Are you sure that's a real spell?" said the girl. "Well, it's not very good, is it? I've tried a few simple spells just for practice and it's all worked for me. Nobody in my family's magic at all, it was ever such a surprise when I got my letter, but I was ever so pleased, of course..."
— "Harry Potter and the Sorcerer's Stone", chapter 6, "The Journey from Platform Nine and Three-Quarters"
Malfoy wasn't present at that time, but if she saw no reason not to talk about her parents, he could easily have heard her mention it at any time — perhaps even that same day on the Hogwarts Express as she's going around trying to help Neville find his toad.
Draco Malfoy would probably use that insult on anyone who doesn't come from a well reputed ancient and noble wizarding family. There are very few such families, and Draco might have researched all of them, so he can confidently guess and call anyone else a mudblood.
Alternately, he might have noticed that Hermione didn't get owl post from her parents during dinners in the common room.
The Malfoy family is a pure-blood family, and similar to many royal families and the upper strata of social culture (for good and bad) - everyone knows everyone. Think of the social interactions of The Hamptons or Beverly Hills, Mumbai, Cambridge, etc.
Draco probably would not have done much research (or needed to be told) because he would have grown up surrounded by those 'worthy' of associating with. And just from the portrayal of the character I doubt Draco would have put that much effort in researching before slinging an insult at anyone, especially one of Harry & Co.
For snobs like the Malfoys, the only thing that would've counted was being part of the Sacred Twenty-Eight. This is a directory of pureblood families, compiled in the early 1930s, by one Cantankerous Nott.
A British real world equivalent would be A Genealogical and Heraldic Dictionary of the Peerage and Baronetage of the British Empire, better known as Burke's Peerage.
Directories such as this exist in several European countries. For instance, The Netherlands have the Nederland's Adelsboek and the Nederland's Patriciaat, Germany has the Genealogisches Handbuch des Adels.
Assuming that pure-blood is modelled at least in part after blue blood, or nobility, Draco wouldn't even need to look up Hermione in any directory. He would've known from growing up in that environment who the pure-blood and 'pure enough'-blood were.
And after all, the wizarding community isn't that big. So anyone not already known to be a witch or a wizard, must be muggle born.
Also, as indicated in this answer, Hermione made no effort to hide it either.
I don't have the books to hand at the moment. I am however relatively certain that in the diagon alley scenes Hermione's parents are around. Something along the line of Arthur Weasley talking to them, after some research i found this on the wikia.
http://harrypotter.wikia.com/wiki/Arthur_Weasley
A few days later, the family, as well as Harry, went to Diagon Alley to purchase school supplies for the upcoming year. While there, they visited Flourish and Blotts where Gilderoy Lockhart was having a book signing event, during which, Arthur got into a fist fight with Lucius Malfoy over Arthur's fascination with Muggles. He was very excited to meet Hermione Granger's parents, both Muggle dentists, insisting on taking them out for a drink.
By this confrontation draco could easily recognise Hermiones parents as muggles.
We know he is explicitly told in Chamber of Secrets.
’‘Though if his school marks don’t pick up,’ said Mr Malfoy, more coldly still, ‘that may indeed be all he is fit for.’‘It’s not my fault,’ retorted Draco. ‘The teachers all have favourites , that Hermione Granger –’‘I would have thought you’d be ashamed that a girl of no wizard family beat you in every exam,’ snapped Mr Malfoy.
This takes place during Harry's accidental trip to Diagon Alley.
As other answers have pointed out she makes no point of hiding it.
Also the wizarding world is really a small tight-knit community and Hogwarts isn't really a huge school (okay the building's huge but the student population isn't....its probably less than 300 students). Information (gossip) probably travels quite quickly through the whole of the community. And given "pure-bloods" are all related they likely all know each-other and probably most "half-bloods" at least by reputation.
I don't have anything in the text to support this, but I didn't want to leave it as a comment: I think it's as simple as Malfoy not recognizing Hermione's last name. If she were not Muggle-born, Malfoy surely would be familiar with the Granger family (or her mother's family, if only her mother were magical). The entire Malfoy family is obsessed with bloodlines, so I've always assumed they just know, from memory, which families are Muggle-born and which are not.
Quite simply, she doesn't have a last name associated with any (living) wizarding lineages. There are a few known Granger wizards, but not for at least a couple of centuries. Potter and Weasley were, no matter their reputation, magical at the root.
That said, it is likely her association with Harry that caused him to single her out for this insult.
Putting it simply, Draco was just being a brat when he called Hermione a mud-blood. He had no way of knowing if she was or wasn't, but all pure-bloods believe lesser witches/wizards are mud-blood. Not to mention, pure-blood families make a point of staying inter-connected to other pure-blood families, or at least being in the know of them. I'm sure it was just matter of the Granger name never coming up that he knew.
