exult vs cheer up

exult and cheer up both are verbs.

Word	Noun	Adjective	Verb	Adverb
exult	No	No	Yes	No
cheer up	No	No	Yes	No

As verbs, cheer up is a hypernym of exult; that is, cheer up is a word with a broader meaning than exult:

Other hypernyms of exult include cheer, chirk up.

exult (verb)	cheer up (verb)
to express great joy	become cheerful
feel extreme happiness or elation	cause (somebody) to feel happier or more cheerful