Computing $DF$ when $F(x, y)=f(x, y, g(x, y))$

Question

I am trying to compute the derivative of $F$ in terms of $Df$ and $Dg$ given the formula $F(x, y)=f(x, y, g(x, y))$. But I don't know how to apply chain rule to this.

Answer 1

HINT: $$dF=\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy +\frac{\partial f}{\partial g(x,y)}d[g(x,y)]$$ and $$d[g(x,y)]=dg=\frac{\partial g}{\partial x}dx +\frac{\partial g}{\partial y}dy$$ Hence $$dF=\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy +\frac{\partial f}{\partial g}\cdot \left(\frac{\partial g}{\partial x}dx +\frac{\partial g}{\partial y}dy\right)$$

And the rest is for you.