Intuition explaining $\frac{\partial^2 f}{\partial x \partial y}$

Question

This is probably something I should already know, but I would like to know an intuitive way to think about the partial x, partial y derivative of $f$. I understand that $\frac{\partial ^2 f}{\partial x^2} $ can be understood as the concavity in the x-direction. Is there a similar way of thinking about $\frac{\partial^2 f}{\partial x \partial y}$?

Answer 1

Among the easiest functions you know (i. e. polynomials) can you construct one for which this derivative is nonzero and all other derivatives (also of higher and lower order) areally zero?

Yes, $ xy $.

This polynomial should then be pretty representative of what the derivative $\partial_{xy} $ means. Best look at a graph of $ xy $. All I can say with words is that the graph has two different curvatures, it is convex along one diagonal and concave along the other.

The two dimensional Taylor polynomial of degree 2 supports the claim that that polynomial represents the meaning of $\partial_{xy} $ even for other functions