Find ${\partial \over \partial x}(zyx^2)$
So I was discussing with one of my students that you can keep track of partial derivatives with the following:
$$\begin{align} {\partial \over \partial x} (zyx^2) &= zy{\partial \over \partial x}(x^2) \tag{$zy$ is constant} \\ &=2xyz\end{align}$$
My colleague told me that teaching students this way is bad because it is abuse of notation.
I don't fully understand why this might be abuse of notation exactly. Is there something mathematically incorrect that could come up with using such a method to explain the partial derivative?
If $y$ and $z$ are each independent of $x$, then it is not an abuse of notation and your colleague is wrong.
If either of $y$ or $z$ depend on $x$, then your answer is wrong.