Conservation of Energy in Mechanics for Point Mass

[Bucho]

Reading "Atmospheric Thermodynamics" I'm stumped almost as soon as I've started. I've probably bitten off more than I can chew and this also might even be more of a math question than a physics one but where I'm stuck is where they "simplify" from:

mv . dv/dt = -mgv . e_z (where e_z is a unit vector on the z-axis and the dots signify scalar multiplication by v)

to

d/dt 1/2mv . v = d/dt 1/2 mv² = -mg dz/dtWhat I do get is that on the right we have v . e_z = dz/dt, since velocity is the derivative of displacement (z) with respect to time.

What I don't get is the operation on the left by which the d/dt is just magically pulled out, leaving the v behind to work its own kind of magic on the other v. The text simply says "This equation can be simplified:" so I guess the authors presume a reader of more skill than I currently have in terms of playing with derivatives. What is it about a derivative that allows that d/dt to be stripped of its v and yanked out in front like that? And where the heck does that 1/2 appear from?

 

[matteo137]

It is just the derivative
$$\dfrac{dv^2}{dt} = 2 v \dfrac{dv}{dt}$$
Which comes from the rule
$$\dfrac{d f(t)^\alpha}{dt} = \alpha f(t)^{\alpha-1} \dfrac{df(t)}{dt}$$

 

[Bucho]

Thanks Matteo, that's somewhat familiar but I'm so rusty on this stuff it's not funny.