Answer
$$\frac{dy}{dt}=\frac{\cos\sqrt{(1+\sqrt t)}}{\sqrt t\sqrt{1+\sqrt t}}$$
Work Step by Step
$$\frac{dy}{dt}=\frac{d}{dt}4\sin\sqrt{1+\sqrt t}=\frac{d}{dt}4\sin(1+t^{1/2})^{1/2}$$
Following the Chain Rule: $$\frac{dy}{dt}=4\cos(1+t^{1/2})^{1/2}\frac{d}{dt}(1+t^{1/2})^{1/2}$$
$$\frac{dy}{dt}=4\cos(1+t^{1/2})^{1/2}\frac{1}{2}(1+t^{1/2})^{-1/2}\frac{d}{dt}(1+t^{1/2})$$
$$\frac{dy}{dt}=2\cos(1+t^{1/2})^{1/2}(1+t^{1/2})^{-1/2}\Big(\frac{1}{2}t^{-1/2}\Big)$$
$$\frac{dy}{dt}=\cos(1+t^{1/2})^{1/2}(1+t^{1/2})^{-1/2}t^{-1/2}$$
$$\frac{dy}{dt}=\frac{\cos\sqrt{(1+\sqrt t)}}{\sqrt t\sqrt{1+\sqrt t}}$$
