ILet $f$ be of class $C^{(2)}$ and let $\displaystyle F(x,y)=f(x,xy)$, then I want to find the mixed partial derivative $\displaystyle F_{12}$.
Here I am letting $g^{1}(x,y)=x$ and $g^{2}(x,y)=xy$. Using the chain rule I get, $F_{1}=f_{1}g_{1}^{1}+f_{2}g_{1}^{2}=f_{1}\cdot 1+f_{2}\cdot y.$ Then I don't know how to find $F_{12}$? Please make a suggestion!