$\begin{align}  & \text{ }y={{({{e}^{x}}\sin x)}^{2}} \\ & \text{let }u={{e}^{x}}\sin x \\ & y={{u}^{2}},\text{ }\frac{dy}{du}=2u \\ & \frac{du}{dx}={{e}^{x}}\frac{d}{dx}(\sin x)+\sin x\frac{d}{dx}({{e}^{x}}) \\ & \frac{du}{dx}={{e}^{x}}\cos x+{{e}^{x}}\sin x \\ & \frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}=2u({{e}^{x}}\cos x+{{e}^{x}}\sin x) \\ & \frac{dy}{dx}=2{{e}^{x}}\sin x({{e}^{x}}\cos x+{{e}^{x}}\sin x) \\ & \frac{dy}{dx}=2{{e}^{2x}}\sin x(\cos x+\sin x) \\\end{align}$