Answer
$$ f'(\theta)= e^{\theta}\cos( e^{\theta}) .$$
Work Step by Step
Recall that $(e^x)'=e^x$
Recall that $(\sin x)'=\cos x$.
Since we have
$$ f(\theta)= \sin(e^{\theta})$$
then the derivative $ f'(\theta)$, using the chain rule, is given by
$$ f'(\theta)=\cos( e^{\theta}) (e^{\theta})'=e^{\theta}\cos( e^{\theta}) .$$
