Newton-Raphson method

At the current iterate $x_k$, the tangent line to the curve $y=f(x)$ is

The tangent is the best linear approximation of $f$ near $x_k$, so we take as next iterate the point where the tangent vanishes ($y=0$):

Write $f$ via the Fundamental Theorem of Calculus from $x_k$ and approximate the integral with a rectangle (constant integrand $f'(x_k)$, the same idea as in the Euler derivation):

Evaluate at $x=\alpha$, where $f(\alpha)=0$, and solve for $\alpha$:

That approximation of $\alpha$ is the next iterate. Approximating the integral with richer quadratures (trapezoid, midpoint, Simpson) produces higher-order methods along this same route.

Let $\alpha$ be a simple root ($f(\alpha)=0$, $f'(\alpha)\ne 0$), $e_k=x_k-\alpha$ the error and $c_2=\frac{1}{2}\frac{f''(\alpha)}{f'(\alpha)}$. Expand $f(x_k)$ by Taylor around $\alpha$; the constant term $f(\alpha)$ vanishes:

Expand the derivative as well:

Divide both expansions. The factor $f'(\alpha)$ cancels and, using $\frac{1}{1+2c_2e_k}=1-2c_2e_k+\mathcal{O}(e_k^2)$:

Subtract $\alpha$ in Newton's formula and substitute: the linear terms in $e_k$ cancel and the quadratic error equation remains. Newton's method has order $p=2$.