How does Newton's Method work?

Question

Before I am told, I want to clarify that I searched first, and I don't believe this to be a repost. I understand the formula in terms of how to apply it, and I've seen graphical representations and everything. I get that we are finding where the tangent line has a root, then choosing a new $f(x)$ at that point and finding the root of its tangent line, effectively closing the distance between x and the root r.

What I do not understand, is what $\frac{f(x)}{f'(x)}$ is actually doing. I know it can be used to find the root x, as it is derived from $y=mx+b$, but how is dividing $f(x)$ by its derivative getting me the root? Why does this work? My intuition is telling me (before I actually tried it) that I was getting some y value, then seeing how many times the slope goes into it; but this would give me the $x$ coordinate, wouldn't it? I can use it, but it's not clicking as to why, and I'd like to fix that so I can actually understand what is going on.

Answer 1

To understand the method, replace the function $f(x)$ by its first order Taylor expansion

$f_L(x)=f(x_0)+f'(x_0)(x-x_0)$

at the point $x_0$ and then calculate the point where this function is zero. Then you will see from where the term $f(x)/f'(x)$ is coming.

Answer 2

When you are at $x_0$ which is "close" to the root, you can expand the function as a truncated Taylor series, that is to say $$f(x)=f(x_0)+(x-x_0) f'(x_0)+O\left((x-x_0)^2\right)$$ So, locally, $$f(x)\approx f(x_0)+(x-x_0) f'(x_0)=0$$ from which the update $$x=x_0-\frac{f(x_0)}{f'(x_0)}$$ and repeat until convergence.