How to obtain partial derivatives of R,G,B

Hi!!! I'm trying to do an edge detector but how I do to each color band to obtain partial derivatives rx,ry, gx, gy, bx, by (for red,green,blue). With these color derivatives I must create a S matrix, where S matrix is rx^2 + gx^2 +bx^2, rxry + gxgy + bxby, rxry + gxgy + bxby, ry^2 + gy^2 +by^2 could you tell me what size have derivatives ? Ok, now , from S, I must calcolate the trace, the eigenvalues and the crrisponding eigenvectors. How I can do all ? thanks you for our time and for answer me