Algorithms for solving f(x) = 0

For a differentiable function $f$, describe numerical algorithms for finding a root of $f(x)=0$.

Bisection, Newton-Raphson and the secant method. Bisection halves a sign-change bracket $[a,b]$ each step: guaranteed but only linear convergence. Newton uses $x_{n+1}=x_n-f(x_n)/f'(x_n)$, converging quadratically near a simple root but may diverge from a poor start. The secant method replaces $f'$ by a finite difference of the last two iterates (order about $1.62$); hybrids such as Brent's method combine safety and speed.