Confusion over re-writing a differential

Question

I'm sure I learned how to do this, but it's not making any sense to me.

$$ v\frac{dv}{ds}=\frac{d}{ds}\left( \frac{v^2}{2}\right) \\ \text{with: v=speed and s=distance}$$

I don't understand how they can be set equal

Answer 1

If we use the chain rule and start from the right hand side, you know that the derivative of $\frac{v^2}{2}$ is $\frac{2v}{2}$, but then we have to multiply by $\frac{dv}{ds}$ because of chain rule.

Answer 2

Or put another way, $\frac{dv^2}{ds}$ = $\frac{dv^2}{dv} \frac{dv}{ds}$ = $2v\frac{dv}{ds}$