Deriving an expression of kinetic energy

Question

How is it that

$$ mv\frac{dv}{dt} = \frac{d}{dt}\left(\frac{1}{2}mv^2\right) $$

I cannot seem to arrive at the expression, getting stuck at $$ \frac{d(mv^2)}{dt}=\frac{dm}{dt}v^2+2mv\frac{dv}{dt} $$

Answer 1

Because $v$ is a function of $t$, you have: $$\frac{d}{dt}v^{2} = 2v \frac{dv}{dt}$$ and thus: $$\frac{d}{dt}\frac{1}{2}mv^{2} = mv \frac{dv}{dt}.$$

Note: Your first expression is true when the mass does not depend on time.