Thank you Bill!

You are quite a huge helper!

Hope you have successes for your future careers as well!

Thanks to Nasser as well!

To be honest, my final step towards the problem will be having the simplest expression of the

result that I can use in practice. For this reason, something like including the expression like

sigma (for summation not for standard deviation) or the most convenient way to understand the

result will be quite useful for making the extenuating practical test for this result.

The final result that I want to make into the simplest expression to be used in practice and extended useages

will be generated from the following codes:

as = Variance[Table[u[i, t] - u[i, t - 1], {t, 1, 3}, {i, 1, 2}]];

aa = Total[Flatten[FullSimplify[as, Element[u[_, _], Reals]]]];

cv = Covariance[Table[u[i, t] - u[i, t - 1], {t, 1, 3}, {i, 1, 2}]];

m = FullSimplify[cv, Element[u[_, _], Reals]];

bb = Total[Flatten[FullSimplify[m - IdentityMatrix[2]*Diagonal]]];

D[Simplify[((aa + 2*bb)^(1/2))*F, Element[u[_, _], Reals]], u[1, 2]]

-> here, the u[1,2] in the last part of the last line can vary. This is the reason why I am trying to derive

a generalized expression for future usages.

Is there a way to do so?

Again..Thank you very much for you sincere helps and hope all successes for your future endeavors!