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

Using termwise division gives $LaTeX: \displaystyle f(x) = - 2 x^{\frac{1}{4}} - x^{\frac{1}{2}} + 4$ . Now the power rule for derivatives can be used instead of the quotient rule. This gives $LaTeX: f'(x) = - \frac{1}{2 \sqrt{x}} - \frac{1}{2 x^{\frac{3}{4}}}$