Answer
$\ln t(\ln t\cos t+\frac{2\sin t}{t})$
Work Step by Step
$\frac{d}{dt}(\ln t)^2\sin t$
$=(\ln t)^2\frac{d}{dt}\sin t+\sin t\frac{d}{dt}(\ln t)^2$
$=(\ln t)^2\cos t+\sin t*2\ln t\frac{d}{dt}\ln t$
$=(\ln t)^2\cos t+2\sin t\ln t*\frac{1}{t}$
$=\ln t(\ln t\cos t+\frac{2\sin t}{t})$
