So I have a very unique application of chain rule (I've been told it's chain rule) that I've never seen before and I really need explained. It relates to a physics problem, but the question is purely mathematical.
Here's the setup. Let z = h(x, y), x = f(y, z), y = g(x)
Then (I've been told) the operator $$\frac{\partial}{\partial x}$$ is given by the following expression: $$\frac{\partial}{\partial x}=\frac{\partial y}{\partial x}\frac{\partial}{\partial y}+\frac{\partial z}{\partial x}\frac{\partial}{\partial z}$$ Which is definitely not the chain rule I learned in Calc III.
In a single-variable context, I think it would be:
Let y = f(x), x = g(y) Then, $$\frac{d}{dy}=\frac{dx}{dy}\frac{d}{dx}$$
Is there a theorem or proof to this that someone can point me to? I've never learned about this situation before in all of my years of calculus, and it's crucial for a physics problem I'm working on.
Thanks so much!