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}<br />

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!