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

Using termwise division gives $LaTeX: \displaystyle f(x) = 5 x^{\frac{1}{4}} + 4 x^{\frac{1}{2}} + 5$ . Now the power rule for derivatives can be used instead of the quotient rule. This gives $LaTeX: f'(x) = \frac{2}{\sqrt{x}} + \frac{5}{4 x^{\frac{3}{4}}}$