Find the derivative $LaTeX: \displaystyle f(x) = \frac{- 4 x^{\frac{3}{2}} + x^{\frac{1}{2}} + 2 x}{x^{\frac{1}{2}}}$

Using termwise division gives $LaTeX: \displaystyle f(x) = 2 x^{\frac{1}{2}} - 4 x + 1$ . Now the power rule for derivatives can be used instead of the quotient rule. This gives $LaTeX: f'(x) = -4 + \frac{1}{\sqrt{x}}$