Harmonic Stage Platform Vibration (Entertainment Engineering)

An entertainment robotics engineer is designing an automated dynamic stage lift for a Las Vegas theatrical production. During a scenic transition, the vertical displacement $y(t)$ of the platform (in meters, for time $t \ge 0$ seconds) exhibits coupled mechanical oscillation described by:

1. State which differentiation rule governs the calculation of the vertical velocity $v(t) = y'(t)$, and identify the component parts.
2. Differentiate $y(t)$ using the Product Rule and trigonometric derivative rules.
3. Compute the exact velocity of the stage platform at time $t = \pi\text{ seconds}$.