Rewrite the equation as
[tex]y = (x-1) (x-2) x^{-1/2}[/tex]
Then by the product rule, the derivative is
[tex]\dfrac{dy}{dx} = (x-2) x^{-1/2} + (x-1) x^{-1/2} - \dfrac12 (x-1) (x-2) x^{-3/2}[/tex]
and we can factorize this as
[tex]\dfrac{dy}{dx} = \dfrac12 x^{-3/2} \left(2 (x-2) x^{3/2-1/2} + 2 (x-1) x^{3/2-1/2} - (x-1) (x-2)\right)[/tex]
[tex]\dfrac{dy}{dx} = \dfrac12 x^{-3/2} \left(2 (x-2) x + 2 (x-1) x - (x-1) (x-2)\right)[/tex]
[tex]\dfrac{dy}{dx} = \dfrac12 x^{-3/2} (3x^2 - 3x - 2)[/tex]
[tex]\dfrac{dy}{dx} = \dfrac{3x^2 - 3x - 2}{2x^{3/2}}[/tex]
and optionally expanded once more (if only to match the provided "Ans") to
[tex]\dfrac{dy}{dx} = \dfrac32 x^{2-3/2} - \dfrac32 x^{1-3/2} - x^{-3/2}[/tex]
[tex]\dfrac{dy}{dx} = \dfrac32 x^{1/2} - \dfrac32 x^{-1/2} - x^{-3/2}[/tex]
[tex]\dfrac{dy}{dx} = \dfrac32 \sqrt x - \dfrac3{2\sqrt x} - \dfrac1{\sqrt{x^3}}[/tex]
