Higher Order Derivatives

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.

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).

Solution 1:
$\begin{eqnarray}
(f_x)_x&=&\frac{\partial}{\partial x}f_x=\frac{\partial}{\partial x}(6x-14xy^3)=6-14y^3\\\\
(f_x)_y&=&\frac{\partial}{\partial y}f_x=\frac{\partial}{\partial y}(6x-14xy^3)=0-42xy^2=-42xy^2\\\\
(f_y)_x&=&\frac{\partial}{\partial x}f_y=\frac{\partial}{\partial x}(12y^2-21x^2y^2)=0-42xy^2=-42xy^2\\\\
(f_y)_y&=&\frac{\partial}{\partial y}f_y=\frac{\partial}{\partial y}(12y^2-21x^2y^2)=24y-42x^2y\\\\
\end{eqnarray}$

Higher partial derivatives and Clairaut's theorem are explained in the following video.