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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!

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  • $\begingroup$ Why do you think this is wrong? If you have a function of one variable, the only possible partial derivative is the ordinary derivative. $\endgroup$ – amd Feb 6 at 20:32
  • $\begingroup$ Thank you for your response. I guess my understanding of the chain rule is perhaps a little basic. I understand the chain rule expression as being valid as normally you can cancel the denominator in the left hand fraction with the numerator in the right hand fraction (so in my example normal dX cancels with partial dX), to get back to the original derivative on the left hand side of the expression. In my example above, I didn't think it was valid to cancel a partial dX with a normal dX. Does that make sense? $\endgroup$ – rm1310 Feb 6 at 21:16

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