Computing $F_{12}$ when $F(x,y)=f(x,xy);F(x,y)=f(x,y,g(x,y))$

Question

I've a question as follows:

Compute $F_{12}$ when $F(x,y)=f(x,xy);F(x,y)=f(x,y,g(x,y))$

I think it to be an application of implicit function theorem . but don't know how to apply,
kindly help with this...