Answer
$$ f'(t)= \frac{1}{2} \frac{1}{\sqrt{t}} e^{\sqrt{t}} .$$
Work Step by Step
Recall that $(e^x)'=e^x$
Since we have
$$ f(t)= e^{\sqrt{t}}$$
then the derivative $ f'(t)$, using the chain rule, is given by
$$ f'(t)=e^{\sqrt{t}} (\sqrt{t})'=\frac{1}{2} \frac{1}{\sqrt{t}} e^{\sqrt{t}} .$$
