im working on image stacks, and I need to calculate second order partial derivatiatives of it. I already know how to calculate derivative on x and y axis, using coefficients calculator: https://web.media.mit.edu/~crtaylor/calculator.html
So I already know how to calculate derivatives on x and y axes, but I also need to get formulas how to calculate first and second order partial derivatives for f'xy and f''xy. I know that I could simply calculate derivatives on x axis first, then calculate from it on y axis, but I would like to obtain simplified formula that uses x and y pixels in image to calculate f'xy/f''xy. Could anyone explain me how to obtain them?
I found somewhere on the web the formula for second order f''xy derivative for images that looks like that:
Would like also to get an explanation whether its correct, and how its acquired