besmirch vs badmouth

besmirch and badmouth both are verbs.

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

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

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

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

besmirch (verb)	badmouth (verb)
charge falsely or with malicious intent; attack the good name and reputation of someone	speak unfavorably about
smear so as to make dirty or stained