Answer
$$2\sec^22\theta e^{\tan 2\theta }$$
Work Step by Step
Given
$$h(\theta )=e^{\tan 2\theta }$$
Then
\begin{align*}
h'(\theta)&=\frac{d}{d\theta}\left(e^{\tan 2\theta }\right) \\
&=e^{\tan 2\theta }\frac{d}{d\theta}\left( {\tan 2\theta }\right) \\
&=2\sec^22\theta e^{\tan 2\theta }
\end{align*}
