6. The displacement, $y(\mathrm{~m})$, of a body in damped oscillation is $y=2 e^{-t} \sin 3 t$. The task is to: a) Use the Product Rule to find an equation for the velocity of the object if $v=\frac{d y}{d t}$

Step 1 ：Given the displacement equation: \(y(t) = 2e^{-t} \sin{3t}\)

Step 2 ：Let \(u(t) = 2e^{-t}\) and \(v(t) = \sin{3t}\)

Step 3 ：Differentiate \(u(t)\) with respect to \(t\): \(u'(t) = -2e^{-t}\)

Step 4 ：Differentiate \(v(t)\) with respect to \(t\): \(v'(t) = 3\cos{3t}\)

Step 5 ：Apply the Product Rule: \(v(t) = u'(t) \cdot v(t) + u(t) \cdot v'(t)\)

Step 6 ：Substitute the derivatives: \(v(t) = -2e^{-t} \sin{3t} + 6e^{-t} \cos{3t}\)

Step 7 ：\(\boxed{v(t) = -2e^{-t} \sin{3t} + 6e^{-t} \cos{3t}}\)