Newton's method
Newton's method uses the derivative of a differentiable function to approximate its real or complex roots. For a function 
, the approximations are defined recursively by ![\[x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)}.\]](http://latex.artofproblemsolving.com/f/0/8/f08ade3a5da34655c6135480dd3285c872dfbc0a.png)
To begin the recursion, an initial guess 
must be chosen. Often the choice of determines which of several possible roots is found, and in some cases the initial guess can cause the recursion to cycle or diverge instead of converging to a root.
Derivation
At each step of the recursion, we have 
and seek a root of . Since all nonconstant linear functions have exactly one root, as long as 
we can construct a tangent-line approximation to and find its root as an approximation. The tangent-line approximation of about is ![\[f(x_i) + (x - x_i)f'(x_i).\]](http://latex.artofproblemsolving.com/c/0/1/c018d0d103a8035c1665a7a2952a4aca6988f004.png)
We seek the value 
such that the above expression equals 
; hence, ![\[f(x_i) + (x_{i+1} - x_i)f'(x_i) = 0.\]](http://latex.artofproblemsolving.com/1/d/d/1dd525c8949bd7cb6301ade781a35dd1844a5573.png)
Dividing by 
(as long as ), ![\[\frac{f(x_i)}{f'(x_i)} + x_{i+1} - x_i = 0.\]](http://latex.artofproblemsolving.com/a/d/e/ade3d00dbdc11182f6f6c7c90671cea015702ace.png)
Therefore, the desired value of 
is ![\[x_{i+1} = x_i - \frac{f(x_i)}{f'(x_{i})}.\]](http://latex.artofproblemsolving.com/5/1/1/5114f7bf2eb289e6a2a9b504f6f3d548d24d83ed.png)
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