Notation for partial derivatives I thought that the meaning of
$$
\frac{\partial f(x, y, z)}{\partial x}
$$
is differentiation on $x$ with fixed $y$ and $z$. So $(x, y, z)$ in the numerator is just saying which variables are fixed. If I need to indicate where the derivative is evaluated, I write it in the right of a vertical bar as a subscript. But today my teacher used $(x, y, z)$ in the numerator to denote where the derivative is evaluated. So, for example,
$$
\frac{\partial f(0, 0, 0)}{\partial x}
$$
means
$$
\frac{\partial f(x, y, z)}{\partial x} \bigg\rvert_{x=0,y=0,z=0}
$$
Is that a standard convention? If so, what is the meaning of
this?
$$
\frac{\partial f(x, y, g(x, y))}{\partial x}
$$
I have two candidates. One is a partial derivative of the composition of $f$ and $g$ where $g$ has some fixed value, and the other is the partial derivative of $f$ on $x$ evaluated at $(x, y, g(x, y))$. I think the two are not the same.

The first interpretation is not correct, since there is no way to give $g$ some fixed value if it depends on $x$. The second interpretation is correct but worded in a confusing way, so let me clarify.
Consider the function $h(x,y)=f(x,y,g(x,y))$. Then
$$
\frac{\partial f(x,y,g(x,y))}{\partial x}=\frac{\partial h(x,y)}{\partial x}.
$$
The result is a function of two variables, call it $k(x,y)$. To calculate e.g. $k(x,5)$ you would compute
$$
k(x,5)=\frac{d}{dx}f(x,5,g(x,5)).
$$
This is a derivative in the usual sense, not a partial derivative any more. By the chain rule, we can calculate $k(x,5)$ as follows:
$$
k(x,5)=f_x(x,5,g(x,5)) + f_z(x,5,g_x(x,5)).
$$
There is nothing special about the number $5$, it could be anything. So replacing $5$ with $y$, we get the formula
$$
\frac{\partial f(x,y,g(x,y))}{\partial x}=f_x(x,y,g(x,y)) + f_z(x,y,g_x(x,y)).
$$
