Definition of a functions with respect to partials

Question

I am stuck with the following problem:

I am given that $$F(x,y)=f(x,y,g(x,y)) =0.$$

I am asked to show $D_1g$ and $D_2g$ with respect to the partials of $f$

My idea was to write that $DF=DfDg$ $\Rightarrow$ $Dg=\frac{DF}{Df}$ but I can't proceed from this step. Any hint would be appreciated!

Thanks in advance!

Answer 1

Notice that $z = g(x,y)$. We apply the chain rule because the third coordinate $z$ is also a function of $x$ and $y$. Differentiating with respect to $x$ gives

$$\frac{\partial F}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial z} \frac{\partial g}{\partial x} = 0,$$

therefore

$$\frac{\partial g}{\partial x} = - \frac{ \frac{\partial f}{\partial x} }{ \frac{\partial f}{\partial z}}.$$

The same method applies to $y$.