Can someone please help me understand What is the difference between によって、次第、and 応じて.
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によって is spoken.
次第 is written very formal. I almost never get to use this when it's meaning is +- that of によって but i use it all the time to describe a sequence (in sequence, in order, in turn) particularly when I'm explaining CPUs to customers.
応じて written, not as formal. As well as 'depends on/depending on' it can also mean in proportion to. so 「天気によって・・・」and 「天気に応じて・・・」are kinda like "depends on the weather" and "weather appropriate"