Difference between ∂w/∂x and ∂f/∂x 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?
 A: 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.
