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Say I had a problem like this:

Prove that the nth derivative of x*e^(x) is (x+n)*e^(x) for all integer n.

Can I use reverse induction to prove for negative n? For example...

Say I proved it for my base case, n=0. In this case, the proof is trivial.

Then I prove that if the nth derivative is (x+n)e^(x), then the (n+1)th derivative is (x+n+1)e^(x). (I didn't provide the proof because there's a similar homework problem here, and the proof is easy anyway.

Can I then use reverse induction to prove that if the nth derivative is (x+n)e^(x), then the (n-1)th derivative is (x+n-1)e^(x), thus extending this case to negative derivatives (i.e., integrals)?

Am I even making sense?

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# Reverse Induction?

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