University Calculus: Early Transcendentals (3rd Edition)

$$y'=\sin t+t\cos t$$
$$y=t\log_3(e^{(\sin t)(\ln3)})$$ We have $\log_3(e^{(\sin t)(\ln3)})=\log_3(e^{\ln3})^{\sin t}=\log_3(3)^{\sin t}=\sin t$ Therefore, $$y=t\sin t$$ The derivative of $y$ is: $$y'=\sin t+t(\sin t)'$$ $$y'=\sin t+t\cos t$$