While studying a practice problem for a classical mechanics class, I encountered the following expressions in the solution given by my instructor:
Having derived an equation for the upward vertical motion of a projectile (including drag) with gravitational acceleration g, velocity v, & proportionality constant k: $$\frac{dv}{dt}=-g-kv^2$$ My instructor then takes the following steps: $$v\frac{dv}{dt}=-(g+kv^2)v$$ $$\frac{1}{2}\frac{dv^2}{dt}=-(g+kv^2)\frac{dy}{dt}$$
My confusion is centered on the left-hand side of the last expression; I've never seen a variable seemingly "absorbed" into a differential before. Is there a rule or theorem for this, or am I missing something obvious?
(Many thanks in advance for any help you can offer!)