In what way is the Newton-Raphson method more beneficial than the bisection method?

The Newton-Raphson method offers several advantages over the bisection method, particularly in the context of computational physics, where the efficiency and accuracy of numerical root-finding techniques are of critical importance. Both methods are used to find the roots of equations, but their mechanisms and performance characteristics differ significantly.

The Newton-Raphson method is based on using the first derivative of a function to estimate the next approximation of the root. It follows the iterative formula:
$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
This method assumes that the function is differentiable in the neighborhood of the root and uses the tangent at a point to find a better approximation. In contrast, the bisection method is a bracketing method that requires two initial points $a$ and $b$ ___ ______ _________ _______ _____ ____.
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