I'm working on some mechanics at the moment and can across this issue in a question.
Distance differentiated in regards to time becomes velocity: $dx/dt$. Differentiate velocity in regards to time and you get acceleration: $d^2x/dt^2$.
If I have $v^2$ and want to differentiate in regards to time, how would you go about doing so? Do I first write velocity as a differential, and how would I even do so $(dx/dt)^2$?
Any help would be much appreciated.
To find $\frac d{dt}\left(v^2\right)$ you use the chain rule $$\frac d{dt}\left(v^2\right)=2v\frac d{dt}v=2va$$ You can certainly write $v^2=\left(\frac {dx}{dt}\right)^2$ but that is not needed here.