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

Using termwise division gives $LaTeX: \displaystyle f(x) = - 4 x^{2} + x + 4$ . Now the power rule for derivatives can be used instead of the quotient rule. This gives LaTeX: f'(x) = 1 - 8 x