I have come across the following in some lecture notes and do not understand the interchange of partial derivatives and normal derivatives, by which I mean $\partial$ and $d$, respectively. I am not sure if the notes are incorrect, or I am merely missing something. Note I am an experimental physicist by trade and so might be missing something obvious, but my reading around in other related posts has just confused me further! Any help would be greatly appreciated.
The notes are as follows -
The function in question is:
$g\left(\frac{x}{w(z)}\right)$,
and we need the derivative of this with respect to $x$.
Defining:
$ X = \frac{x}{w(z)} $
then we can write:
$\frac{\partial g}{\partial x} = \frac{d g}{d X} \frac{\partial X}{\partial x} = \frac{1}{w}\frac{dg}{dX}$.
My question is - Is it $\textit{allowed}$ to have partial and normal derivatives used together like this in the chain rule? It feels slightly wrong to do so! If so, why? If not, what would be the mathematically $\textit{correct}$ way to write this. Thanks!