Find \( \frac{dy}{dt} \) for \( y=\ln (2e^{-t} \sin t ) \)

\( y=\ln (2e^{-t} \sin t )= \ln (2) - t + \ln (\sin t) \)

\( \frac{dy}{dt}=0-1+\frac{\cos t}{\sin t}=-1+\cot (t) \)