About multipart names, Prof. Paul Brians of Washington State University says:
In many European languages family names are often preceded by a preposition (de, da, di, von, and van all mean “of”), an article (le and la mean “the”) or both (du, des, del, de la, della and van der all mean “of the”).
In their original languages the two parts of the name are usually separated by a space, and the prefixed preposition or article is not capitalized unless it begins a sentence. If you take a college course involving famous European names you will be expected to follow this pattern. It’s not “De Beauvoir” but “de Beauvoir”; not “Van Gogh” but “van Gogh.” The only exception is when the name begins a sentence:
- De Gaulle led the Free French
- Charles de Gaulle had a big nose.
Quoting Mark Stevens, the director of general reference at Merriam-Webster’s, Grammarphobia says:
In the names of Frenchmen and -women, de and d’ are almost always lowercased; treatment of du varies. La and Le are almost always capitalized.
With Guillaume de l'Hôpital, the difficulty is whether to mention the surname including or omitting de. The internet uses more than one variant at the beginning of a sentence, but most sites omit the de, and capitalise L:
Since all mathematicians speak about L'Hôpital's Rule, also spelled as L'Hospital's rule, I would go for L'Hôpital, whether it is at the beginning or in the middle of the sentence.