slander vs badmouth

slander is a noun but badmouth is not a noun.

Word	Noun	Adjective	Verb	Adverb
slander	Yes	No	Yes	No
badmouth	No	No	Yes	No

As verbs, badmouth is a hyponym of slander; that is, badmouth is a word with a more specific, narrower meaning than slander:

slander: charge falsely or with malicious intent; attack the good name and reputation of someone
badmouth: speak unfavorably about

Other hyponyms of slander include assassinate, libel, drag through the mud, malign, traduce.

slander (noun)	badmouth (noun)
an abusive attack on a person's character or good name
words falsely spoken that damage the reputation of another

slander (verb)	badmouth (verb)
charge falsely or with malicious intent; attack the good name and reputation of someone	speak unfavorably about