I'm currently studying the fixed point iterative method, and I am confused as to how the derivative of g(x) at a point between a guess and a root can tell us if the method will converge or diverge for this point. For example, for
$f(x) = e^x - 10x = 0$,
$x = g(x)$, where $g(x) = (1/10)e^x$.
When you rearrange it with the root solutions, you will get
$x_{i+1} − x_{r} = {\dfrac{dg}{dx}}(ξ) (x_{i} − x_{r})$,
where $x_{i+1}$ and $x_{i}$ are the guesses of the root, $x_{r}$ is the root, and the mean value theorem is used, with $ξ$ being a value inbetween $x_{i}$ and $x_{r}$.
I understand mathematically, the ${\dfrac{dg}{dx}}(ξ)$ term has to be below 1 for convergence to occur, but I do not understand the intuition behind it.
Does this mean the fixed point method only works for roots where the derivative near a point is relatively flat?
A visual interpretation of the derivative is that we graph the function on the $xy$-plane, then "zoom in" near one point until the curve looks almost like a straight line. The derivative is the slope of this line which the curve looks like. So for simpler intuition, what happens if a piece of the function's graph actually is a straight line segment?
Suppose $g(x) = mx+b$, with $m \neq 1$. The unique point where $g(x)=x$ is $x_r = \frac{b}{1-m}$.
But if we ignore the simple algebraic solution and just try iteration from some guess, we'll have some sequence $(x_i)$ where $x_{i+1} = g(x_i)$. Is this sequence converging to, or getting farther from, the value $x_r$?
$$ \begin{align*} x_{i+1} - x_r &= g(x_i) - x_r \\ &= mx_i + b - x_r \\ &= m(x_i - x_r) + m x_r + b - x_r \\ &= m(x_i - x_r) + (m-1) x_r + b \\ x_{i+1} - x_r &= m(x_i - x_r) \end{align*} $$
So the absolute value of the error in the sequence decreases if $|m|<1$, or increases if $|m|>1$.
Intuitively then, once the sequence is "close enough" to the correct point that the derivative is a "good enough" description of the overall behavior of the iterative algorithm, it will converge if $|g'(x_r)| < 1$, since the derivative approximates a linear slope in a neighborhood of the point. If $|g'(x_r)| > 1$, the sequence can't converge unless it happens to land on $x_N = x_r$ exactly, since for a point $x$ close to $x_r$, the iteration will send the next point farther from $x_r$. If the derivative doesn't exist, or if $g'(x_r) = \pm 1$, we need to look for other means of analyzing things.
Getting away from intuition to precise statements, using the mean value theorem's value $\xi$ is part of a way of proving something general about actual values which aren't "arbitrarily close to $x_r$" (since no such values really exist in ordinary real analysis).