The time derivative of kinetic energy

[LagrangeEuler]

Lets consider $$T(\vec{p})=\frac{\vec{p}^2}{2m}=\frac{\vec{p}\cdot \vec{p}}{2m}$$. Then $$\frac{dT}{dt}=\vec{v}\cdot \vec{F}$$.
And if we consider
$$T=\frac{p^2}{2m}$$ than $$\frac{dT}{dt}=\frac{1}{2m}2p\frac{dp}{dt}$$
Could I see from that somehow that this is $$\vec{v}\cdot \vec{F}$$?

 

[PeroK]

Lets consider $$T(\vec{p})=\frac{\vec{p}^2}{2m}=\frac{\vec{p}\cdot \vec{p}}{2m}$$. Then $$\frac{dT}{dt}=\vec{v}\cdot \vec{F}$$.
And if we consider
$$T=\frac{p^2}{2m}$$ than $$\frac{dT}{dt}=\frac{1}{2m}2p\frac{dp}{dt}$$
Could I see from that somehow that this is $$\vec{v}\cdot \vec{F}$$?

Well, $\vec {p} = m \vec {v}$ and $\vec F = \frac{d\vec p}{dt}$ is Newton's second law.