Restoring Quadratic Convergence of Newton's Method

I am studying for finals and am looking over previous problems that I missed significant points on. My instructor doesn't comment on why things are incorrect and he doesn't discuss homework, so I am on my own to figure out why I am wrong. Anyway, I would like some help figuring out where I went wrong.

Prove that if is a zero of multiplicity of the function , then quadratic convergence in Newton's iteration will be restored by making this modification

**Proof:**

If is a zero of multiplicity then can be rewritten as

where and so we have Writing as its Taylor series expansion about

where are between and Thus we have

This is where I went wrong. I originally thought that the derivative was just , but depend on so that is not true.

This is suppose to go to zero, but I don't really see how. I suppose it has something to do with .

Any help is appreciated, also if you have a different way to do this I will be glad to see it. Thank you in advance.