I'll link the original question here: The complex version of the chain rule
I've tried now for an extended period of time to understand the following proof; however, I think it's time to ask for help.
C. Dubussy proved the result as below in the linked post:
$$\frac{\partial}{\partial x} (g \circ f) = (\frac{\partial g}{\partial x} \circ f)(\operatorname{Re}\frac{\partial f}{\partial x})+(\frac{\partial g}{\partial y} \circ f)(\operatorname{Im}\frac{\partial f}{\partial x})$$ and $$\frac{\partial}{\partial y} (g \circ f) = (\frac{\partial g}{\partial x} \circ f)(\operatorname{Re}\frac{\partial f}{\partial y})+(\frac{\partial g}{\partial y} \circ f)(\operatorname{Im}\frac{\partial f}{\partial y}).$$
Now, I can write out the Jacobian of $g \circ f$ and compare it with the Jacobian that arises from $J(g \circ f)*J(f)$ and get the right identities above; however, when I try to think about the above partials individually I am left uncertain in my derivation.
My Work:
Let's just focus on the $\frac{\partial}{\partial x} (g \circ f) $ term since the identities are symmetric in some sense.
Let $f= u+ iv$, $g = s+it$ where $u,v,s,t$ are all real functions in $x,y$.
$$\frac{\partial}{\partial x} (g \circ f) = \frac{\partial}{\partial x}s(u,v) +i \frac{\partial}{\partial x}t(u,v) = \frac{\partial s}{\partial u } \frac{\partial u}{\partial x} + \frac{\partial s}{\partial v } \frac{\partial v}{\partial x} + i (\frac{\partial t}{\partial u } \frac{\partial u}{\partial x} + \frac{\partial t}{\partial v } \frac{\partial v}{\partial x}) $$
Am I right in then concluding:
$$\bigg(\frac{\partial s}{\partial u } \frac{\partial u}{\partial x} + i\frac{\partial t}{\partial u } \frac{\partial u}{\partial x}\bigg) + \bigg(\frac{\partial s}{\partial v } \frac{\partial v}{\partial x} + i\frac{\partial s}{\partial v } \frac{\partial v}{\partial x}\bigg) $$
$$\bigg(\frac{\partial (s+it)}{\partial u } \bigg)\frac{\partial u}{\partial x} + \bigg(\frac{\partial (s+it)}{\partial v } \bigg)\frac{\partial v}{\partial x} = (\frac{\partial g}{\partial x} \circ f)(\operatorname{Re}\frac{\partial f}{\partial x})+(\frac{\partial g}{\partial y} \circ f)(\operatorname{Im}\frac{\partial f}{\partial x}) $$
I suppose I feel uneasy working with the chain rule in multiple dimensions since it has rarely come up in my studies, and I just want to verify the final step I've presented here.
Thanks in advance.
