I see that, but I do not understand how that is done algebraically,
I get $\displaystyle \frac{1}{2}\sqrt{-x}(-1)-1$
$\displaystyle \frac{-1\sqrt{-x}}{2}-1$
$\displaystyle -\frac{\sqrt{-x}}{2}-1$
Which is as far as I get when trying to figure out how it becomes $\displaystyle \frac{-1}{2\sqrt{-x}}-1$
Maybe $\displaystyle \frac{x}{2\sqrt{-x}}-1$ ? closer, but where does the factor of x go?
The problem is NOT the differentiation. I suspect that you did not tell us the whole problem! After differentiating they set the derivative equal to 0 and then started solving for x. That would be a natural thing to do if the problem were not just to find the derivative but to find maximum and minimum values for the function.