What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.

Step 1
Mixed Derivative theorem:" If the function f(x,y) and its partial derivatives ${\displaystyle f_x,f_y,f_{xy}\text{and}{f}_{yx}}$ are all defined in any open interval (a,b) and all are continues in the interval, then ${\displaystyle f_{xy}(a,b)=f_{yx}(a,b)}$".
That is, mixed derivative theorem says that the mixed partial derivatives are equal.
Thus, there is no need of calculating all the mixed partial derivatives. Only one case is enough.
Step 2
For example consider the function ${\displaystyle f(x,y)=x^3{y}^3+x^{2}y+y^{2}x}$.
Find the first order partial derivatives as follows.
${\displaystyle \frac{\partial }{\partial x}(x^3{y}^3+x^{2}y+y^{2}x)=\frac{\partial }{\partial x}\left(x^3{y}^3\right)+\frac{\partial }{\partial x}\left(x^{2}y\right)+\frac{\partial }{\partial x}\left(y^{2}x\right)}$
${\displaystyle =3y^3{x}^2+2yx+y^2}$
${\displaystyle \frac{\partial }{\partial y}(x^3{y}^3+x^{2}y+y^{2}x)=\frac{\partial }{\partial y}\left(x^3{y}^3\right)+\frac{\partial }{\partial y}\left(x^{2}y\right)+\frac{\partial }{\partial y}\left(y^{2}x\right)}$
${\displaystyle =3x^3{y}^2+x^2+2xy}$
Step 3
Now, find the mixed partial derivatives as,
${\displaystyle \frac{{\partial }^2}{\partial y\partial x}(x^3{y}^3+x^{2}y+y^{2}x)=\frac{\partial }{\partial x}(3x^3{y}^2+x^2+2xy)}$
${\displaystyle =9y^2{x}^2+2x+2y}$
${\displaystyle \frac{{\partial }^2}{\partial x\partial y}(x^3{y}^3+x^{2}y+y^{2}x)=\frac{\partial }{\partial y}(3y^3{x}^2+2yx+y^2)}$
${\displaystyle =9x^2{y}^2+2x+2y}$
That is, ${\displaystyle \frac{{\partial }^{2}f}{\partial y\partial x}=\frac{{\partial }^{2}f}{\partial x\partial y}}$.