#### Answer

1) To prove the formula,
- First, write the formula of $a(t)$ as the rate of change of $v(t)$.
- Then write the formula of $v(t)$ as the rate of change of $s(t)$.
- Finally, use the Chain Rule with $ds$ to modify the first formula, then use the second formula to change it into the formula that needs to be proved.
2) $\frac{dv}{dt}$ means the rate of change of the velocity of the particle according to time.
$\frac{dv}{ds}$ means the rate of change of the velocity of the particle according to displacement, or position.

#### Work Step by Step

1) Show that $$a(t)=v(t)\frac{dv}{ds}$$
We know that the acceleration $a(t)$ of a particle is the rate of change of its velocity $v(t)$.
Therefore, $$a(t)=\frac{dv}{dt}\hspace{0.5cm}(1)$$
However, we also know that the velocity $v(t)$ of a particle is the rate of change of its displacement $s(t)$.
Therefore, $$v(t)=\frac{ds}{dt}\hspace{0.5cm}(2)$$
So, we would use the Chain Rule to write $(1)$ into $$a(t)=\frac{dv}{ds}\frac{ds}{dt}$$ $$a(t)=\frac{dv}{ds}v(t)$$
The formula is proven.
2) $\frac{dv}{dt}$ means the rate of change of the velocity of the particle according to time. That means, at time $t$, velocity changes at a $a_t$ value, which is the acceleration of the particle at that time $t$ $a(t)$.
$\frac{dv}{ds}$ means the rate of change of the velocity of the particle according to displacement, or position. That means, at position $s$, velocity was changing at an $a_s$ value, which is the acceleration of the particle at that position $s$ $a(s)$.