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## Higher Order DerivativesSince $f_x$ and $f_y$ are functions of $x$ and $y$, we can take derivatives of these functions to get second derivatives. There are four such second derivatives, since each time we can differentiate with respect to $x$ or $y$.
The main result about higher derivatives is:
A consequence of this theorem is that we don't need to keep track of the order in which we take derivatives. We just need to keep track of how many times we differentiate with respect to each variable. Higher partial derivatives and Clairaut's theorem are explained in the following video. |