Question: Newtons-Raphsons method is easy to implement in Mathematica but in Matlab it seems a bit difficult. I don't get if I can pass a function to a function and how to use the derivative as a function.

```
newtonRaphson[f_, n_, guess_] := 
 If[n == 0, guess, newtonRaphson[f, n - 1, guess - f[guess]/f'[guess]]]
newtonRaphsonOptimize[f_, n_, guess_] := 
 If[n == 0, guess, 
  newtonRaphsonOptimize[f, n - 1, guess - f'[guess]/f''[guess]]]

```

It doesn't seem like you can derive neither function-handles nor functions defined in a file but I might be wrong.

Answer: 
There's no way to *algebraically* take derivatives of function handles or functions defined in m-files. You would have to do this *numerically* by evaluating the function at a number of points and approximating the derivative.

What you're probably wanting to do is [differentiation of symbolic equations](http://www.mathworks.com/help/toolbox/symbolic/f1-82523.html), and you need the [Symbolic Math Toolbox](http://www.mathworks.com/products/symbolic/) for that. Here's an example of finding a root using the [Newton-Raphson method](http://en.wikipedia.org/wiki/Newton%27s_method):

```
>> syms x            %# Create a symbolic variable x
>> f = (x-4)^2-4;    %# Create a function of x to find a root of
>> xRoot = 1;        %# Initial guess for the root
>> g = x-f/diff(f);  %# Create a Newton-Raphson approximation function
>> xRoot = subs(g,'x',xRoot)  %# Evaluate the function at the initial guess

xRoot =

    1.8333

>> xRoot = subs(g,'x',xRoot)  %# Evaluate the function at the refined guess

xRoot =

    1.9936

>> xRoot = subs(g,'x',xRoot)  %# Evaluate the function at the refined guess

xRoot =

    2.0000

```

You can see that the value of `xRoot` comes close to the value of the true root (which is 2) after just a couple of iterations. You could also place the function evaluation in a while loop with a condition that checks how big a difference there is between each new guess and the previous guess, stopping when that difference is sufficiently small (i.e. the root has been found):

```
xRoot = 1;                     %# Initial guess
xNew = subs(g,'x',xRoot);      %# Refined guess
while abs(xNew-xRoot) > 1e-10  %# Loop while they differ by more than 1e-10
  xRoot = xNew;                %# Update the old guess
  xNew = subs(g,'x',xRoot);    %# Update the new guess
end
xRoot = xNew;                  %# Update the final value for the root

```
```
% Friday June 07 by Ehsan Behnam.
% b) Newton's method implemented in MATLAB.
% INPUT:1) "fx" is the equation string of the interest. The user 
% may input any string but it should be constructable as a "sym" object. 
% 2) x0 is the initial point.
% 3) intrvl is the interval of interest to find the roots.
% returns "rt" a vector containing all of the roots for eq = 0
% on the given interval and also the number of iterations to
% find these roots. This may be useful to find out the convergence rate
% and to compare with other methods (e.g. Bisection method).
%
function [rt iter_arr] = newton_raphson(fx, x, intrvl)
n_seeds = 10; %number of initial guesses!
x0 = linspace(intrvl(1), intrvl(2), n_seeds);
rt = zeros(1, n_seeds);

% An array that keeps the number of required iterations.
iter_arr = zeros(1, n_seeds);
n_rt = 0;

% Since sometimes we may not converge "max_iter" is set.
max_iter = 100;

% A threshold for distinguishing roots coming from different seeds. 
thresh = 0.001;

for i = 1:length(x0)
    iter = 0;
    eq = sym(fx);
    max_error = 10^(-12);
    df = diff(eq);
    err = Inf;
    x_this = x0(i);
    while (abs(err) > max_error)
        iter = iter + 1;
        x_prev = x_this;

        % Iterative process for solving the equation.
        x_this = x_prev - subs(fx, x, x_prev) / subs(df, x, x_prev);
        err = subs(fx, x, x_this);
        if (iter >= max_iter)
            break;
        end
    end
    if (abs(err) < max_error)
        % Many guesses will result in the same root.
        % So we check if the found root is new
        isNew = true;
        if (x_this >= intrvl(1) && x_this <= intrvl(2))
            for j = 1:n_rt
                if (abs(x_this - rt(j)) < thresh)
                    isNew = false;
                    break;
                end
            end
            if (isNew)
                n_rt = n_rt + 1;
                rt(n_rt) = x_this;
                iter_arr(n_rt) = iter;
            end
        end
    end        
end
rt(n_rt + 1:end) = [];
iter_arr(n_rt + 1:end) = [];

```