For the following sets of variables, find all the relevant second derivatives. In all cases, first find general expressions for the second derivatives and then substitute variables at the last step. f(x,y)=x^2y-xy^2,where x = st and y = s/t

Step 1
Given function is ${\displaystyle f(x,y)=x^{2}y-x{y}^2\text{ where }x=st,y=s}$.
The all relevant second derivatives are ${\displaystyle f_{xx},f_{yy},f_{xy},f_{yx}}$.
Step 2
Find partial derivative with respect to x for the function ${\displaystyle f(x,y)=x^{2}y-x{y}^2}$.
${\displaystyle \Rightarrow f_x=2xy-y^2}$
Find partial derivative with respect to y for the function ${\displaystyle f(x,y)=x^{2}y-x{y}^2}$.
${\displaystyle \Rightarrow f_y=x^2-2xy}$
Differentiate function ${\displaystyle f_x=2xy-y^2}$ withe respect to x to get ${\displaystyle f_{\times }}$.
${\displaystyle \Rightarrow f_{xx}=2y}$
Substitute ${\displaystyle y=\frac{s}{t}\in f_{xx}=2y}$.
${\displaystyle \Rightarrow f_{xx}=\frac{2s}{t}}$
Differentiate function ${\displaystyle f_y=x^2-2xy}$ with respect to y to get ${\displaystyle f_{yy}}$.
${\displaystyle \Rightarrow f_{yy}=-2x}$
Substitute x=st in ${\displaystyle f_{yy}=-2x}$.
${\displaystyle \Rightarrow f_{yy}=-2st}$
Differentiate function ${\displaystyle f_y=x^2-2xy}$ with respect to x to get ${\displaystyle f_{xy}}$.
${\displaystyle \Rightarrow f_{xy}=2x-2y}$
Substitute ${\displaystyle x=st,y=s\in f_{xy}=2x-2y}$.
${\displaystyle \Rightarrow f_{xy}=2st-\frac{2s}{t}}$
Differentiate function ${\displaystyle fx=2xy-y^2}$ with respect to y to get ${\displaystyle f_{yx}}$.
${\displaystyle \Rightarrow f_{yx}=2x-2y}$
Substitute ${\displaystyle x=st,y=s\in f_{yx}=2x-2y}$
${\displaystyle \Rightarrow f_{yx}=2st-\frac{2s}{t}}$.
Step 3
Therefore, all relevant second derivatives of function

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