mis- (1)youdaoicibaDictYouDict[mis- 词源字典]
prefix meaning "bad, wrong," from Old English mis-, from Proto-Germanic *missa- "divergent, astray" (cognates: Old Frisian and Old Saxon mis-, Middle Dutch misse-, Old High German missa-, German miß-, Old Norse mis-, Gothic missa-), perhaps literally "in a changed manner," and with a root sense of "difference, change" (compare Gothic misso "mutually"), and thus from PIE *mit-to-, from root *mei- (1) "to change" (see mutable); see Watkins.

Others [Barnhart] see in Proto-Germanic *missa- the stem of an ancient past participle, related to Old English missan "fail to hit" (see miss (v.)), which is from the same PIE root.

Productive as word-forming element in Old English (as in mislæran "to give bad advice, teach amiss"). In 14c.-16c. in a few verbs its sense began to be felt as "unfavorably" and was used as an intensive prefix with verbs already expressing negative feeling (as in misdoubt). Practically a separate word in Old and early Middle English (and often written as such). Old English also had an adjective (mislic "diverse, unlike, various") and an adverb (mislice "in various directions, wrongly, astray") derived from it, corresponding to German misslich (adj.).[mis- etymology, mis- origin, 英语词源]
mis- (2)youdaoicibaDictYouDict
in mischief, miscreant, etc., represents Old French mes- "bad, badly, wrong, wrongly," from Vulgar Latin minus-, from Latin minus "less" (see minus), which was not used as a prefix. Perhaps influenced in Old French by *miss-, the Frankish equivalent of mis- (1).