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The full derivation is arigatō < arigatau < arigataku < arigatashi < ari + katashi.
Arigatō is a euphonic form of arigataku occurring when the medial -k- drops out. This results in arigataku > arigatau. /au/ regularly becomes a long IPA: [oː] via regular phonological rules.
Arigataku is an adverbial conjugation of the adjective arigatashi (modern arigatai). Typically a verb will follow the adverbial form as is the case here. In modern Japanese, arigatō is considered an abbreviation for arigatō gozaimasu. Historically sōrō, gozaru, and others are attested.
The adjective arigatashi is composed of two other words: the verb ari "to be" and the adjective katashi "difficult". The initial consonant in katashi becomes voiced (/k/ -> /g/) due to the compounding of the two separate words. The original meaning was "difficult to be", ie "rare" and thus "special", worthy of being thankful for.
Early usage of the word arigatō begins to appear during the early 15th century. The adjective arigatashi, from which the word derives, appears in the oldest literature of the 8th century.
There is absolutely no linguistic relationship to the Portuguese word obrigado of the same meaning.
- 有り難う, 有難う: thank you