What are some "derivative notations" worth to memorize? Like $\frac{x''}{x'}$ is equal to $\left(\ln\left(x'\right)\right)'$ (x depends on t)

Question

It can make the calculation easier especially in differential equations but I only know this and $-\frac{x'}{x^2}$ = $\left(\frac{1}{x}\right)'$

Answer 1

A few that immediately come to mind are: $$\begin{gather} ydx+xdy = d(xy) \\ \frac{ydx-xdy}{y^2} = d\left(\frac xy\right) \\ \frac{ydx+xdy}{xy} = d(\ln(xy)) \\ \frac{ydx-xdy}{xy} = d\left(\ln\left(\frac xy\right)\right) \\ \frac{ydx+xdy}{1+x^2y^2} = d(\arctan (xy)) \end{gather}$$