Board Thread:Theories and Speculation/@comment-27717552-20160513151229/@comment-734582-20160609113409

ThrowingHands wrote: Kakashi didn't need to try (and fail) to add nature change to the Rasengan to invent the Chidori. Knowledge of the Rasengan didn't help Kakashi create the Chidori. That part is actually never made clear. Rasengan is about gathering chakra in your hand and rotating it, chidori gathers chakra in the hand and forms it to lightning. Shape change is minimally used on this for precision. Whether such a jutsu can be invented without knowing Rasengan is never elaborated.

The Wiki doesn't officially list it as a derived jutsu, so it's best to say that they're related.