The rule is
[tex]\frac{d}{dx}x^n = nx^{n-1}[/tex]
which is the power rule. You pull down the exponent to place it as the coefficient. So that explains the 1/2 pull out front. Then we subtract 1 from the exponent.
The expression you wrote can be simplified or rewritten like this
[tex]\frac{d}{dx}x^n = nx^{n-1}\\\\\frac{d}{dx}\left[x^{1/2}\right] = \frac{1}{2}x^{\frac{1}{2}-1}\\\\\frac{d}{dx}\left[x^{1/2}\right] = \frac{1}{2}x^{-\frac{1}{2}}\\\\\frac{d}{dx}\left[x^{1/2}\right] = \frac{1}{2}\frac{1}{x^{\frac{1}{2}}}\\\\\frac{d}{dx}\left[x^{1/2}\right] = \frac{1}{2x^{\frac{1}{2}}}\\\\\frac{d}{dx}\left[x^{1/2}\right] = \frac{1}{2\sqrt{x}}}\\\\[/tex]
Optionally, we can multiply top and bottom by [tex]\sqrt{x}[/tex] to rationalize the denominator.
