If we have $w = f(x, y, z)$ and $z = g(x, y)$, how would I take the partial derivative of $w$ with respect to $x$?
Would it be $\frac{\partial w}{\partial x} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial x}$? But then what is the difference between $\frac{\partial w}{\partial x}$ and $\frac{\partial f}{\partial x}$? Are they not the same thing like they are in calc 1?
When you write \begin{align*} \dfrac{\partial w}{\partial x}=\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial z}\cdot\dfrac{\partial z}{\partial x}, \end{align*} you are dealing with the function \begin{align*} \varphi:(x,y)\rightarrow f(x,y,g(x,y)). \end{align*} A better expression would be \begin{align*} \dfrac{\partial\varphi}{\partial x}=\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial z}\cdot\dfrac{\partial g}{\partial x}, \end{align*} or even better that \begin{align*} \dfrac{\partial\varphi}{\partial x}=D_{1}f+D_{3}f\cdot\dfrac{\partial g}{\partial x}. \end{align*}
Meanwhile, when you write \begin{align*} \dfrac{\partial w}{\partial x}=\dfrac{\partial f}{\partial x}, \end{align*} you are dealing with the function \begin{align*} (x,y,z)\rightarrow f(x,y,z) \end{align*} and you have labelled this function as $w$.
In short, there is some sort of composite of functions in the first one (to the third coordinate of $w$, or $f$) and the second one is merely the function $w$ (or $f$), so in two cases you are dealing with entirely two different functions.