# Thread: Proof of derivatives through induction

1. ## Proof of derivatives through induction

Hey guys,

Have a problem that was set as exam revision and was told that there'd be a question similar on the exam, so I'm very keen on understanding how to do this problem well.

The problem asks:

Let f(x) = {e-1/x^2, if x =/= 0
{0, if x = 0

Prove that f(n)(0) = 0 for positive integer values of n.

I've figured that induction would be a good way to go about solving this problem obviously, although I'm not too sure how to prove it for any case.
Any help would be greatly appreciated

Thanks in advance,
Mark

2. ## Re: Proof of derivatives through induction

You need the fact that $\displaystyle \lim_{t\to\infty}e^{-t^2}t^n=0$ for all n.

Prove by induction on n that there exists a polynomial $\displaystyle p_n$ such that $\displaystyle f^{(n)}(x)=e^{-1/x^2}p_n(1/x)$ for $\displaystyle x\ne 0$ and $\displaystyle f^{(n)}(0) = 0$. The latter part for n + 1 is proved by the definition of the derivative using the induction hypothesis.