Once, to never forget, the derviation. Given a point $x_i$ called the
\emph{initial estimate}, and a function $f(x)$, we can make an
approximation $x_u$ for $x_0$ s.t. $f(x_0)=0$ by constructing the
$1$-st order approximation to $f$ at $x_i$. The equation of this
line, translated to $t = x - x_i$ is $$l(t) = f(x_i) + t f'(x_i).$$
Solving for $t_u$ s.t. $l(t_u) = 0$ gives $t_u = - f(x_i)/f'(x_i)$, or
$$x_u = u(x_i) = x_i - {f(x_i) \over f'(x_i)}.$$ The function $u$ can
now be iteratively applied to refine the estimate.