Assume that $r(t)$ and $s(t)$ are two functions in $t$. When is the following true?
$$ \frac{\partial r}{\partial t} \left(\frac{\partial s}{\partial t}\right)^{-1} = \frac{\frac{\partial r}{\partial t}}{\frac{\partial s}{\partial t}} = \frac{\partial r}{\partial s} $$
Would you have a reference? Please let me know if the question has already been asked.
If $ r=r(t)$ and $ s=s(t)$ are functions then according to the chain rule we have $$ \frac {dr}{ds} =\frac {\frac {dr}{dt}}{ \frac {ds}{dt}}$$
For example if $r=t^2$ and $s=\cos t$ then $$ \frac {dr}{ds} =\frac {\frac {dr}{dt}}{ \frac{ds}{dt}}=\frac {2t}{ -\sin t}$$