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Your cancelling is wrong.

Yeah that isn't the derivative of that

....Not that helped any in terms of trying to do this with the limit definition, lol.

lol..

you need chain rule to solve this

Here is also a detailed proof of chain rule
http://kruel.co/math/chainrule.pdf

or rather to be able to break a function into smaller functions

you can probably incorporate the chain rule proof to solve this problem