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Are we going to need to know how to do proofs (ie. showing work) for the midterm or final exam?

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Yes - if an anwer to a problem needs justification (= showing your work), then you will be expected to provide that justification.

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I think I was a little vague. My question intended was: are we expected to show proofs (ie. chain rule) which include the steps to acquiring how the chain rule is comprised? another example of a proof (in the sense of linear algebra) is the steps of showing a+b=b+a (but for calculus). I hope this helps because I have taken some calculus and I had a lot of problems with proofs (ie. the chain rule). Thanks

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You will not be asked to prove theorems in the exams; in particular, you will not need to memorize the proof of the chain rule.