Chapter 3 - Differentiation - 3.3 Product and Quotient Rules - Exercises - Page 122: 22

$$ f'(x)= \frac{27}{2}x^{3/2}+\frac{2}{x^2}.$$

First, we simplify $ f(x) $ as follows $$ f(x)=\frac{9x^{5/2}-2}{x}=9x^{3/2}-2x^{-1}.$$ Now, the derivative is given by $$ f'(x)=9(3/2)x^{1/2}+2x^{-2}=\frac{27}{2}x^{3/2}+\frac{2}{x^2}.$$