Is there a way to study "how good" is the Newton's Raphson method applied to a function? My question is as simple as that. 
When we're applying the fixed point algorithm, we can see if it's going to converge or diverge finding the derivative and checking if the absolute value of that function is less or greater than $1$ on a given interval, and we can estimate if at any given point if the convergence will be fast or slow.
Is there a way to study the "contractivity" of the Newton's Raphson method?
Thanks for your time. 

As a fixed point iteration
$$
N(x)=x-\frac{f(x)}{f'(x)}
$$
has derivative
$$
N'(x)=\frac{f(x)f''(x)}{f'(x)^2}
$$
and the usual estimates apply. Since at the solution $f(x_*)=0$, you can get intervals around $x_*$ with arbitrarily small contraction constants, compatible with the quadratic convergence speed.
