Extension of Clairaut's theorem for partial derivatives

Question

I was wondering if anyone knows of an extension of Clairaut's theorem for interchanging the order of partial differentiation. For example, just recently I noticed that for a lot of functions

$$f_{xyy} = f_{yyx}$$.

Answer 1

I would not really call that an extension. If the first order derivatives of $f$ also satisfy the conditions of Clairaut's theorem, then by two consecutive applications of that theorem

$$f_{xyy}=(f_{xy})_y=(f_{yx})_y=(f_y)_{xy}=(f_y)_{yx}=f_{yyx}$$