$\begin{align}  & y=4{{x}^{2}}\sin x{{\cos }^{2}}x \\ & \frac{dy}{dx}=4{{x}^{2}}\sin x\frac{d}{dx}({{\cos }^{2}}x)+4{{x}^{2}}{{\cos }^{2}}x\frac{d}{dx}(\sin x)+\sin x{{\cos }^{2}}x\frac{d}{dx}(4{{x}^{2}}) \\ & \frac{dy}{dx}=4{{x}^{2}}\sin x(-2\sin x\cos x)+4{{x}^{2}}{{\cos }^{2}}x(\cos x)+\sin x{{\cos }^{2}}x(8x) \\ & \frac{dy}{dx}=-8{{x}^{2}}{{\sin }^{2}}x\cos x+4{{x}^{2}}{{\cos }^{3}}x+8x\sin x{{\cos }^{2}}x \\\end{align}$