Answer
$$\frac{dy}{dt}=-24\cos^3(\sec^23t)\sin(\sec^23t)\sec^23t\tan3t$$
Work Step by Step
$$\frac{dy}{dt}=\frac{d}{dt}\cos^4(\sec^23t)$$
Following the Chain Rule: $$\frac{dy}{dt}=4\cos^3(\sec^23t)\frac{d}{dt}(\cos(\sec^23t))$$
$$\frac{dy}{dt}=4\cos^3(\sec^23t)(-\sin(\sec^23t))\frac{d}{dt}(\sec^23t)$$
$$\frac{dy}{dt}=4\cos^3(\sec^23t)(-\sin(\sec^23t))2\sec3t\frac{d}{dt}(\sec3t)$$
$$\frac{dy}{dt}=8\cos^3(\sec^23t)(-\sin(\sec^23t))\sec3t(\sec3t\tan3t\frac{d}{dt}(3t))$$
$$\frac{dy}{dt}=-24\cos^3(\sec^23t)\sin(\sec^23t)\sec^23t\tan3t$$
