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.

*

*Yes, $\frac{\partial f(x,y,z)}{\partial x}$ is derivative w.r.t. $x$ at fixed $y,z$.

*$\frac{\partial f(0,0,0)}{\partial x}$ is not standard notation. Strictly speaking, it should be zero, because $f(0,0,0)$ is a constant which does not depend on $x$. Sometimes, yes, it is used as a shorthand for $\frac{\partial f(x,y,z)}{\partial x}|_{x=y=z=0}$. But you should only do that if it is extreamly clear from the context what you mean. Generally, avoid this notation.

*The third expression is a derivative w.r.t. $x$, but $x$ appears two times in the numerator. Using standard derivation-rules you can compute
$$\frac{\partial f(x,y,g(x,y))}{\partial x} = \frac{\partial f(x,y,z)}{\partial x}\bigg\rvert_{z=g(x,y)} + \frac{\partial f(x,y,z)}{\partial z}\bigg\rvert_{z=g(x,y)} \cdot \frac{\partial g(x,y)}{\partial x}$$

