Answer
$$\frac{\sin \left(x\right)}{2\sqrt{x}}+\sqrt{x}\cos \left(x\right)$$
Work Step by Step
Given
$$ f(x)=\sqrt{x}\:\sin \:x $$
Since
\begin{aligned}
f'(x)&=\:\frac{d}{dx}\left(\sqrt{x}\:\sin \:x\:\:\right)\\
&=\frac{d}{dx}\left(\sqrt{x}\right)\sin \left(x\right)+\frac{d}{dx}\left(\sin \left(x\right)\right)\sqrt{x}\\
&=\frac{\sin \left(x\right)}{2\sqrt{x}}+\sqrt{x}\cos \left(x\right)
\end{aligned}
We can note that $f(x) $ is decreasing when $f'(x)<0$ and $f(x) $ is increasing when $f'(x)>0 $