Reverse Induction for Proving Negative nth Derivatives of x*e^(x)

[Char. Limit]

Homework Statement

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?

 

[micromass]

Hmm, I'm not sure if this is correct. You do have to use reverse induction though. But isn't it easier to show "if it holds for -n, then it holds for -n-1". Or is this what you meant?

 

[Char. Limit]

Hmm, I'm not sure if this is correct. You do have to use reverse induction though. But isn't it easier to show "if it holds for -n, then it holds for -n-1". Or is this what you meant?

Well, that would probably work too. EDIT: Since my base case is n=0, I don't see much of a difference.