To solve

$$f(x)=0,$$

Newton's method replaces the function near the current estimate $x_n$ by its tangent line. The tangent crosses the $x$-axis at

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.$$

Near a simple root and under suitable smoothness conditions, the error can decrease quadratically: once sufficiently close, the number of correct digits may roughly double at each iteration.

This speed is local rather than guaranteed globally. A poor initial guess, a small or zero derivative, or unfavorable function geometry can cause slow convergence, convergence to a different root, or divergence.

Newton's method therefore trades the robustness of bracketing methods for potentially much faster local convergence and a need for derivative information.