Derivative of a curve's velocity vector

Question

Why is that in the equation below the second derivative of the position vector has a minus in the middle instead of plus after the product rule? I didn't get why we can just replace - sign in front of a if that's the reason.

where;

Answer 1

Just the chain rule.

Note that $\dot{r}(t)=(aC'(t),S'(t))^T$.

Also, $C(t)=\cos\big(\pi-(1-\cos\omega t)\big)$ and $$ C'(t)=-\sin\big(\pi-(1-\cos\omega t)\big)\cdot \pi \sin(\omega t)\cdot\omega =\omega \pi\sin(\omega t)\cdot (-S(t)) $$