Find the derivative $LaTeX: \displaystyle f(x) = \frac{- 4 x^{\frac{5}{2}} + 4 x^{\frac{3}{2}} - 2 x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Using termwise division gives $LaTeX: \displaystyle f(x) = - 4 x^{2} + 4 x - 2$ . Now the power rule for derivatives can be used instead of the quotient rule. This gives LaTeX: f'(x) = 4 - 8 x