Why does v · dv/dt equal 1/2 d(v²)/dt?

[Seung Ju Yoo]

In a book I was reading, it says
F=mv'=P'

so dot producting on both sides with v

F ⋅ v = mv ⋅ dv/dt = 1/2 m d(v²)/dt = d(1/2 m v^2)/dtI really don't get how v ⋅ dv/dt = 1/2 d(v²)/dt.
I have seen few threads and they say it's about product rule, but they don't really explain in detail.

Could anyone help me with this?

 

[PeroK]

In a book I was reading, it says
F=mv'=P'

so dot producting on both sides with v

F ⋅ v = mv ⋅ dv/dt = 1/2 m d(v²)/dt = d(1/2 m v^2)/dtI really don't get how v ⋅ dv/dt = 1/2 d(v²)/dt.
I have seen few threads and they say it's about product rule, but they don't really explain in detail.

Could anyone help me with this?

Welcome to PF!

$v^2 = \textbf{v.v}$

Can you now differentiate that equation?

 

[jedishrfu]

V^2 = V . V

And the time derivative of it is V' . V + V . V'

 

[Seung Ju Yoo]

Welcome to PF!

$v^2 = \textbf{v.v}$

Can you now differentiate that equation?

V^2 = V . V

And the time derivative of it is V' . V + V . V'

Oh.. I see. I did not now that d(x ⋅ y)/dt = x' ⋅ y + x ⋅ y'

Knowing this, going right from left is easy, but I guess going left to right needs some practice to spot!

Thank you both peroK and Jedishrfu!

 

[Chestermiller]

In Cartesian component form, what is $\vec{v}\centerdot d\vec{v}$?

Chet