Suppose I have a function $ f(x, y, g(x, y)) $
How would I express $ \frac{\partial f}{\partial x} $? Using the chain rule, you'd naturally come up with $ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial g} \frac{\partial g}{\partial x} $, except in this expression, $ \frac{\partial f}{\partial x} $ is really only the partial derivative of $f$ with respect to that one parameter, and not $x$. So, my question is, what notation would I use to show this differentiation that is less ambiguous and meaningless than $ \frac{\partial f}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial g} \frac{\partial g}{\partial x} $?
You have a function $f(x,y,z)$ presumably, and then you take a composition $h(x,y) = f(x,y,g(x,y))$. The chain rule here is
$$ \frac{\partial h}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial z}\frac{\partial g}{\partial x} $$
and similarly for $y$. You should verify that on your own, and check a couple examples to convince yourself.