# mixed partial derivative of function of $C^{(2)}$

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!

-
The sub/superscript notation used here is unnecessarily confusing. –  JohnD Dec 16 '12 at 5:38

By $F_{12}$, do you mean $\displaystyle{{\partial^2 F\over \partial x\,\partial y}}$?
If so, take the partial with respect to $y$.
Maybe, you mean $\dfrac{\partial^2{F}}{\partial{x}\partial{y}}$? –  M. Strochyk Dec 16 '12 at 5:35