V dot (dv/dt) = (0.5)*(d/dt)*(v^2) ?

[wavingerwin]

This is not a homework question, but a derivation in my class which I can't get around.

Homework Statement

Step1 $$F=ma$$
Step2 $$\vec{F} = m\frac{d\vec{v}}{dt}$$
Step3: Multiply both side by v $$\vec{F}.\vec{v} = m\vec{v}.\frac{d\vec{v}}{dt}$$
Step4 $$Power = \frac{d}{dt}\frac{1}{2}m\vec{v}.\vec{v}$$
Step5 $$Power = \frac{d}{dt}\frac{1}{2}mv^{2} = \frac{dK}{dt}$$

Homework Equations

The Attempt at a Solution

I just can't figure out where did the $$\frac{1}{2}$$ come from in step 4.

Please help & thanks in advance!

 

[rock.freak667]

v²=v.v

So that d/dt(v²)=2v*dv/dt

They just wrote v² as v.v

 

[wavingerwin]

So that d/dt(v²)=2v*dv/dt

Aha, I see.
Or more elaborately:

$$\frac{d}{dt}v^{2} = \frac{dv}{dt}\frac{d}{dv}v^{2} = \frac{dv}{dt}2v$$

Thanks rock!