First, for context: I am new to both Julia and ForwardDiff, but I am experienced at coding. For my project, I need to compute the first derivative of a 2x2 determinant where the elements of the determinant are second-order partial derivatives; specifically, I need and where

I am not sure why the two mixed partial derivatives don't cancel; it might have something to do with the fact that this is a ternary data system (where x1 + x2 + x3 is constrained to equal 1).
At any rate, is this possible in ForwardDiff, or do I need to do finite differentiation on the sigma obtained from a ForwardDiff hessian?