create a function that performs newton's method without the user inputting the derivative in matlab

In MATLAB, you can create a function that performs Newton's method without the user inputting the derivative by using the diff function to numerically calculate the derivative. Here's an example of how you can do it:

main.m
function root = newtonsMethod(f, x0, epsilon, maxIterations)
    syms x
    f = sym(f);
    fprime = diff(f, x);

    x_i = x0;
    iteration = 0;
    while abs(subs(f, x, x_i)) > epsilon && iteration < maxIterations
        f_i = subs(f, x, x_i);
        fprime_i = subs(fprime, x, x_i);
        x_i = x_i - f_i/fprime_i;
        iteration = iteration + 1;
    end

    if iteration == maxIterations
        disp('Max iterations reached before convergence');
    end

    root = x_i;
end
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Here's how you can use this function:

main.m
f = 'x^2 - 2';
x0 = 1;
epsilon = 1e-6;
maxIterations = 100;

root = newtonsMethod(f, x0, epsilon, maxIterations);
disp(root);
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In this example, the newtonsMethod function takes the function f, an initial guess x0, a tolerance epsilon (to determine the convergence criteria), and a maximum number of iterations maxIterations as input. The function then internally computes the derivative of f using the diff function and iteratively applies Newton's method until either the convergence criteria is met or the maximum number of iterations is reached. The final estimate of the root is returned as the output.