A 'brute force approach' probably fails, so that we try a 'step by step' solution. First step is to perform an integration be part obtaining...
$\displaystyle \int_{0}^{x} \frac{\sin t}{\sqrt{x-t}}\ dt = -2\ |\sin t\ \sqrt{x-t}|_{0}^{x} + 2\ \int_{0}^{x} \cos t\ \sqrt{x-t}\ dt= 2\ \int_{0}^{x} \cos t\ \sqrt{x-t}\ dt$ (1)
From (1) we derive...
$\displaystyle \frac{d}{dx} \int_{0}^{x} \frac{\sin t}{\sqrt{x-t}}\ dt = 2 \frac{d}{dx} \int_{0}^{x} \cos t\ \sqrt{x-t}\ dt= \int_{0}^{x} \frac{\cos t}{\sqrt{x-t}}\ dt$ (2)
Now with a little of patience or using Wolfram if You don't have patience [
...] You find that...
$\displaystyle \int_{0}^{x} \frac{\cos t}{\sqrt{x-t}}\ dt = \sqrt{2 \pi}\ \{ \cos x\ \text{C} (\sqrt{\frac{2 x}{\pi} }) + \sin x\ \text{S} (\sqrt{\frac{2 x}{\pi}})\}$ (3)
… where C and S are the Fresnel Cosine and Fresnel Sine functions, so that, remembering that is $\displaystyle \Gamma(\frac{1}{2})= \sqrt{\pi}$ You obtain finally…
$\displaystyle D^{\frac{1}{2}} \sin x = \sqrt{2}\ \{ \cos x\ \text{C} (\sqrt{\frac{2 x}{\pi} }) + \sin x\ \text{S} (\sqrt{\frac{2 x}{\pi}})\}$ (4)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$