"열심히 공부하다" is a common phrase which means "study hard".
What does 열심히 come from: 열심하다 ?
What 열심히 conjugation and formality is this? Present informal?
1 Answer
-히 is an adverb form (and is thus not conjugated for politeness, formality, or even honorificity, which distinguishes it from verbal nouns ending in -ㅁ or -기).
Same principle as 확실히 (certainly), 천천히 (slowly), 성급히 (hastily), 막연히 (vaguely). It may present as just as -이, especially with stems that end in -ㅅ, e.g. 깨끗이 (cleanly), and other irregular verbs, e.g the famous 빨리 (quickly) comes from 빠르다 + 히.
However, this is not a productive form, meaning that one has to learn which verbs form adverbs this way. There is another way to make verbs (and adjectives / descriptive verbs) into adverbs: with -게, which is the more "productive" one; 깨끗하게 and 빠르게 also exist. For verbs that can take both, there are interesting grammatical reasons why one is preferred over the other for which context, but it is usually easier to learn these case-by-case.
(There is also an even more fossilised adverbial form -오/우, one that is no longer productive at all. This is easiest learnt as a kind of lexical derivation in earlier stages of the language instead: e.g. 넘다 + -오/우 = 너무.)
