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But this is so much incovenient! The convenient would be dA = dx dy.

Let's see now d²A...

d²A = d dA = d²x y + dx dy + dx dy + x d²y

Now dx dy appears! But, is not a convenient expression, because d²x y and x d²y appears too in the equation!

How to solve this problem? We want that A = x y and dA = dx dy. Do you understand this conceptual problem of defition?