I know how to find one of many half derivatives of $\sin(x)$ and $\cos(x)$ which are,
$${D^{\frac{1}{2}}}\sin(x)=\sin(x+\frac{\pi}{4})$$
$${D^{\frac{1}{2}}}\cos(x)=\cos(x+\frac{\pi}{4})$$
Using these is it possible to find a half derivative for $\tan(x)$ by letting.
$$\tan(x)=\frac{\sin(x)}{\cos(x)}$$
Does the Quotient rule apply for half derivatives as well?
For example,
$$D^{\frac{1}{2}}\tan(x)=\frac{D^{\frac{1}{2}}[\sin(x)]\cos(x)-D^{\frac{1}{2}}[\cos(x)]\sin(x)}{\cos^2(x)}$$
As a side question, does the Product and Chain rule also apply for half derivatives, or some other form of it?