2.13 Examples of repeated partial differentiation

Solution. We have $f_{x}=\,{6x^{2}+3y^{2}}$ and $f_{y}=\,{6xy+4y^{3}.}$ Continuing, $f_{xx}=(f_{x})_{x}=\,{12x,}$ $f_{xy}=(f_{x})_{y}=\,{6y,}$ $f_{yx}=(f_{y})_{x}=\,{6y}$ and $f_{yy}=(f_{y})_{y}=\,{6x+12y^{2}.}$ The only non-zero third order partial derivatives are:

Finally, the only non-zero fourth order derivative is $f_{yyyy}=(f_{yyy})_{y}=\,{24.}$