Answer
$4\times \dfrac {1}{\cos ^{2}\left( \theta ^{2}+3\theta +2\right) }\times \left( 2\theta +3\right) $
Work Step by Step
$\dfrac {d}{d\theta }\left( 4\tan \left( \theta ^{2}+3\theta +2\right) \right) =4\times \dfrac {1}{\cos ^{2}\left( \theta ^{2}+3\theta +2\right) }\times \dfrac {d}{d\theta }\left( \theta ^{2}+3\theta +2\right) = 4\times \dfrac {1}{\cos ^{2}\left( \theta ^{2}+3\theta +2\right) }\times \left( 2\theta +3\right) $
