Now, how would the existence of one partial derivative necessarily imply the existence of another? It would be one thing if I could prove that these partial derivatives [math] D_i f_i(x) [/math] themselves were partially differentiable, and then through arguing for some converse of the theorem for equality of mixed partial derivatives that all other jth partial derivatives in n must exist for each function in the set S, but I don't know anything in the toolset that would allow for this