Trying to understand the magic the author of this PDF did explaining the critically damped equation
This is simply the product rule in reverse:

$$\frac{d}{dt}[e^{\alpha t}v_{tr}] = e^{\alpha t}\frac{dv_{tr}}{dt} + v_{tr}\frac{d}{dt}[e^{\alpha t}]
=e^{\alpha t}\frac{dv_{tr}}{dt} + \alpha e^{\alpha t} v_{tr}$$