Explaining derivative being used to derive tensor properties I'm doing some self-study on tensor calculus for physics, and I've come across a derivative that I can't quite wrap my head around. It looks like this:
$\frac{d}{dt}(\frac{\partial x'^i}{\partial x^j})=\frac{\partial^2 x'^i}{\partial x^j\partial x^k}\frac{dx^k}{dt}$
where summation over repeated indices is implied. This came up when investigating a change of coordinates from the unprimed to the primed frame.
Right now, I'm trying to interpret it as this $\frac{\partial x'^i}{\partial x^j}$ being some variable, say $w$, that depends on each $x^k$ and thus we need the chain rule:
$\frac{dw}{dt}=\frac{\partial w}{\partial x^k}\frac{dx^k}{dt}$
But, then I start to get hung up on the notation — is it legal to assert
$\frac{\partial w}{\partial x^k}=\frac{\partial (\frac{\partial x'^i}{\partial x^j})}{\partial x^k}=\frac{\partial^2 x'^i}{\partial x^j\partial x^k}$?
I am wary of manipulating derivatives exactly as I manipulate fractions, because I fear I am sweeping subtle things under the rug.
Can someone verify whether my interpretation here is correct, or offer some other insight?

Your interpretation is correct. You could also see this as an operator equality:
$$\frac{d}{dt} = \frac{\partial x^k}{\partial t}\frac{\partial}{\partial x^k}$$
(which is just another statement of the chain rule) and then use this to replace the $\frac{d}{dt}$ in your equation.
