Since $f_x$ and $f_y$ are functions of $x$ and $y$, we can take derivatives of $f_x$ and $f_y$ to
get second derivatives. There are four such
second derivatives, since each time we can differentiate with
respect to $x$ or $y$. Each of these second derivatives has
multiple notations, and we have listed some of them.

A nice result regarding second partial derivatives is Clairaut's Theorem, which tells us
that the mixed variable partial derivatives are equal.

Clairaut's Theorem
If $f_{xy}$ and $f_{yx}$ are both defined and continuous in
a region containing the point $(a,b)$, then
$$f_{xy}(a,b)=f_{yx}(a,b).$$

A consequence of this theorem is that we don't need to keep
track of the order in which we take derivatives.

Example 1: Let $f(x,y)=3x^2-4y^3-7x^2y^3$.
Previously, we determined that $f_x=6x-14xy^3$ and
$f_y=-12y^2-21x^2y^2$. We have four second derivatives, but
as Clairaut's Theorem tells us, $f_{xy}=f_{yx}$, so we really only
need to compute three of them (we do all four to illustrate the
theorem).