Answer
$$\frac{dy}{dt}=\frac{3}{2}e^{3\sin(t/2)}\cos(t/2)$$
Work Step by Step
$$\frac{dy}{dt}=\frac{d}{dt}\Big(e^{\sin(t/2)}\Big)^3$$
According to the Chain Rule:
$$\frac{dy}{dt}=3\Big(e^{\sin(t/2)}\Big)^2\frac{d}{dt}\Big(e^{\sin(t/2)}\Big)'$$
$$\frac{dy}{dt}=3e^{2\sin(t/2)}e^{\sin(t/2)}\frac{d}{dt}(\sin(t/2))$$
$$\frac{dy}{dt}=3e^{3\sin(t/2)}\cos(t/2)\frac{d}{dt}(t/2)$$
$$\frac{dy}{dt}=\frac{3}{2}e^{3\sin(t/2)}\cos(t/2)$$