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.
[Math] Notation for partial derivatives
calculusnotationpartial derivative
Best Answer
Congratulations, you have met one of the worst ambiguities in mathematical notation!
Assume you have a function of two variables, $f \colon A \times B \to \mathbb{R}$, where $A$ and $B$ are subsets of $\mathbb{R}$. The notation $$\frac{\partial f}{\partial x}(x_0,y_0)$$ is commonly used to denote the value of the partial derivative of $f$ with respect to the first variable, evaluated at $(x_0,y_0)$. This is the cleanest use of the notation for partial derivatives.
Anyway, it sometimes happens to use some lazy piece of notation such as $$\frac{\partial f(x,g(x,y))}{\partial x}$$ to denote the partial derivative of the map $(x,y) \mapsto f(x,g(x,y))$. This is imcompatible (in general) with the interpretation of the same formula as
This is bad, but it seems we have to live with it. Why? Just spend a couple of minutes and think about the second interpretation. To be rigorous, we should have written $$ \frac{\partial}{\partial x} \left( f \circ \left( (x,y) \mapsto (x,g(x,y)) \right) \right) (x,y), $$ which is a true nightmare.