By the product and chain rules, \[\frac{dy}{dx}=\frac32\sec^2\!\left(\frac x2\right)\cos2x-6\tan\!\left(\frac x2\right)\sin2x.\] At \(x=\pi/3\), \(\sec^2(\pi/6)=4/3\), \(\cos(2\pi/3)=-1/2\), \(\tan(\pi/6)=1/\sqrt3\) and \(\sin(2\pi/3)=\sqrt3/2\). Therefore \[\frac{dy}{dx}=\frac32\frac43\left(-\frac12\right)-6\frac1{\sqrt3}\frac{\sqrt3}{2}=-1-3=-4.\]