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I am working on simplifying a differential equation, and I am trying to figure out if a simplification is valid. Specifically, I'm trying to determine if:

[itex]\frac{\del^2 p(x)}{\del p(x) \del x} = \frac{\del^2 p(x)}{\del x \del p(x)}[/itex]

where p(x) is a function of x. Both p(x) and x are assumed to be continuous.

From what I found on wikipedia at http://en.wikipedia.org/wiki/Partial_derivatives (at the bottom of the subsection "Formal definition"), it appears that all partial derivatives have this commutative property if the functions are continuous.

However, a reputable colleague of mine said that this is not the case here. He said that the commutative property doesn't hold since p(x) is being differentiated with respect to itself. He did not have time to explain it thoroughly or give a proof.

Can anyone give a proof (or a strong argument) showing that the commutative property is/is not valid here?

Thank you very much for your help!

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# Commutative property of partial derivatives

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