I have two functions one $f(x,y)=xy$, and the other $g(x,y,z)=e^{x y z}$. How can I calculate $\frac{df(x,y)}{dg(x,y,z)}$? What would be a well defined measure of variation of f with respect to $g$?
I tried using chain rule, however I can see it does not work because $g$ can change due to changes of $z$ with $x, y$ constant and then $f$ will not change.
Regards.
As far as I know the symbols $\frac{\textrm{d}f(x,y)}{\textrm{d}g(x,y,z)}$ do not have any standard meaning. You would need to specify exactly what you meant by this thing, or what you wanted it to do.
One way to make this meaningful would be to define $f,g: \mathbb{R}^3 \to \mathbb{R}$ with $f(x,y,z) = xy$ and $g(x,y,z) = e^{xyz}$. Then you could define
$$\frac{\textrm{d}f}{\textrm{dg}}\big|_{(x,y,z)} (\vec{v}) = \frac{\textrm{d}f(\vec{v})}{\textrm{d}g(\vec{v})}$$.
Not sure if that is what you want though?