partial derivatives commuteThe limit definitions of the first partial derivatives 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If we interate these to get the mixed partial second derivatives in both orders (first wrt x, then wrt y, or vice versa), even before we take the limit, these two independent limit operation expressions agree when we perform them in both orders: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You don't even need Maple to see this, there are just 8 terms and they all cancel in pairs (but are rearranged in order in the two expressions). Maple just makes it quick and easy.\302\267BUT you have to remember that we can interchange the limits only if they both exist. In order to evaluate the second limit, the first derivative must be continuous at the point so that we can difference nearby values! This must be true in both orders, which implies that the two first order partial derivatives are both continous at the point if the two limiting expressions agree.