exhaust vs cough up

exhaust is a noun but cough up is not a noun.

Word	Noun	Adjective	Verb	Adverb
exhaust	Yes	No	Yes	No
cough up	No	No	Yes	No

As verbs, cough up is a hyponym of exhaust; that is, cough up is a word with a more specific, narrower meaning than exhaust:

exhaust (noun)	cough up (noun)
system consisting of the parts of an engine through which burned gases or steam are discharged
gases ejected from an engine as waste products

exhaust (verb)	cough up (verb)
wear out completely	discharge (phlegm or sputum) from the lungs and out of the mouth
eliminate (a substance)	give reluctantly
use up the whole supply of
use up (resources or materials)
deplete