
Proof by Induction
I need some help figuring out the following proof that my professor put in my notes...
Let . Then, . Prove by induction that where is a polynomial.
This is pretty easy to see why it works as we can continue the product rule forever here and continue to get f(x) times a polynomial. However, I'm not sure how to prove it by induction. Any help would be appreciated.

Tis is clearly not true, as in the base step, , and is not a polynomial...


It's false as written for all n>0, but it looks like the following is true:
where is a polynomial.
You should be able to prove this by induction by using the product and quotient rules, and then some algebraic simplification (getting a common denominator). I think you also need to use the fact that the derivative of a polynomial is a polynomial.

I guess there is no real use in trying to guess the original meaning, but it seems to me that the most reasonable one is that , so that , etc.