[tex]f(x)=x(2x-3)^\frac{1}{2}\\\\f'(x)=x'(2x-3)^\frac{1}{2}+x\left[(2x-3)^\frac{1}{2}\right]'=(2x-3)^\frac{1}{2}+x\cdot\frac{1}{2}(2x-3)^{-\frac{1}{2}}\cdot2\\\\=(2x-3)^\frac{1}{2}+x(2x-3)^{-\frac{1}{2}}=(2x-3)^\frac{1}{2}+\dfrac{x}{(2x-3)^\frac{1}{2}}\\\\=\dfrac{(2x-3)^\frac{1}{2}\cdot(2x-3)^\frac{1}{2}+x}{(2x-3)^\frac{1}{2}}=\dfrac{2x-3+x}{\sqrt{2x-3}}=\boxed{\dfrac{3x-3}{\sqrt{2x-3}}}[/tex]
