I am currently learning about the multiple root issue for Newton's method. My textbook hinted at tweaking the function using $$h(x)=\frac{f(x)}{f′(x)}$$ and making the iteration function to actually be $$g(x)= x - \frac{h(x)}{h'(x)}$$
My textbook claims that this method is also quadratically convergent but provided no proof and I am lost as to how it is quadratically convergent. I tried to do some more research but all I could find is this: https://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/iteration%20methods/accelerating%20the%20convergence/mrac.html
Can anybody show me how this quadratically convergent or at least point me in the direction to prove it myself?
Hint: That method is known as Schröder's Method and the iteration formula may be obtained by applying Newton's method to $f(x)/f'(x)$ instead of $f(x)$.