Answer
$(8x-\displaystyle \frac{1}{2\sqrt{x}})(\sqrt{x}-\frac{2}{x^{2}}) +(4x^{2}-\sqrt{x})(\frac{1}{2\sqrt{x}}+\frac{4}{x^{3}})$
Work Step by Step
$(f\cdot g)^{\prime}=f^{\prime}g+fg^{\prime}$
...written in exponential form,
$f(x)=4x^{2}-x^{1/2},\displaystyle \qquad f^{\prime}(x)=8x-\frac{1}{2}x^{-1/2}$
$g(x)=x^{1/2}-2x^{-2},\displaystyle \qquad g^{\prime}(x)=\frac{1}{2}x^{-1/2}+4x^{-3}$
$\displaystyle \frac{dy}{dx}=(f\cdot g)^{\prime}(x)=$
$=(8x-\displaystyle \frac{1}{2}x^{-1/2})(x^{1/2}-2x^{-2}) +4x^{2}-x^{1/2})(\frac{1}{2}x^{-1/2}+4x^{-3})$
...back to radical form...
$=(8x-\displaystyle \frac{1}{2\sqrt{x}})(\sqrt{x}-\frac{2}{x^{2}}) +(4x^{2}-\sqrt{x})(\frac{1}{2\sqrt{x}}+\frac{4}{x^{3}})$
... we do not need to expand the answer..