Answer
$$\frac{dy}{dt}=8\sin2t(1+\cos2t)^{-5}$$
Work Step by Step
$$\frac{dy}{dt}=\frac{d}{dt}(1+\cos2t)^{-4}$$
According to the Chain Rule:
$$\frac{dy}{dt}=-4(1+\cos2t)^{-5}\frac{d}{dt}(1+\cos2t)$$
$$\frac{dy}{dt}=-4(1+\cos2t)^{-5}\Big(0-\sin2t\frac{d}{dt}(2t)\Big)$$
$$\frac{dy}{dt}=-4(1+\cos2t)^{-5}(-2\sin2t)$$
$$\frac{dy}{dt}=8\sin2t(1+\cos2t)^{-5}$$