How calculate the second and third numerical derivative of one variable f(x)

How can write a MATLAB to calculate the second and third numerical derivative of one variable f(x) ??

Answer 1

[Jim Riggs]

The Matlab "diff" function is basically the Backward difference formula. There are much more accurate ways to compute numerical derivatives.

If you like, you can program your own function(s). The central difference calculation is much better than either forward or backward method. Use Forward difference to calculate the derivative at the first point, and backward difference to calculate the derivative at the last point. Everywhere in between, use the central difference formula. This allows you to compute a derivative at every point in your vector, and will provide better results than using recursive applications of "diff".

If you are interested in this method, I can provide equations for even better accuracy than these.

[EDIT] Here is a slightly more accurate version for the Forward/Backward difference:

Source: https://en.wikipedia.org/wiki/Finite_difference_coefficient

Your Matlab function would look somethng like this:

function dy = Nderiv2(y,h)
% Compute the second derivative of input vector y, with spacing h
n = length(y);
for i=1:n
    switch i
        case 1
            % use Forward difference equation for y''
            dy(i) =  ...
        case n
            % use backward difference equation for y''
             dy(i) = ...
        otherwise
            % use central difference equation for y''
            dy(i) = ...
    end
end
end

The third derivative function is similar, except that the central difference formula uses a range on y of y(i-2), y(i-1) y(i+1) y(i+2), so you need two points on either side of y(i) to perform this calculation.

function dy = Nderive3(y,h)
% compute the third derivative of input vector y with spacing h
n = length(y);
for i=1:n;
    switch i
        case {1, 2}
            % use forward difference equation for y'''
            dy(i) = ...
        case {n-1, n}
            % use backward difference equation for y'''
            dy(i) = ...
        otherwise
            % use central difference equation for y'''
            dy(i) = ...
    end
end
end

[Walter Roberson]

Is this what gradient() does already?

[Jim Riggs]

It appears that gradient() is superior to diff() (gradient() uses the central difference method, and returns a vector the same length as the input), however, gradient() only computes the first derivative.

The question is regarding the second and third derivative.

[Guillaume]

"however, gradient() only computes the first derivative."

gradient(gradient(gradient(somevector))) %3rd derivative

Given the lack of details in the question, I don't see why you'd need anything else more complicated.

[Jim Riggs]

You make a good point. (But I personally woud never consider calculating a third derivative this way.)

Answer 2

[Matt J]

Matlab's diff command will do this directly,

https://www.mathworks.com/help/matlab/ref/diff.html