Could someone tell me if I done this problem correctly? I am learning the power rule but having some difficulty remembering some of the algebraic steps.
Yes, that is correct. Well done.
If you wanted, you could have simplified it further to one fraction which would be $\displaystyle = \frac{2x\cos (x)+ \sin (x)}{2\sqrt{x}}$ but your answer is still correct and would be accepted.
This rule is called the 'Product Rule'. $\displaystyle \frac{\mathrm{d}(uv)}{\mathrm{d}x} = \frac{\mathrm{d}(u)}{\mathrm{d}x}v + \frac{\mathrm{d}(v)}{\mathrm{d}x}u$.