Chapter 2 - The Derivative - 2.4 The Product And Quotient Rules - Exercises Set 2.4 - Page 147: 8

$6x + 3 - \frac{27}{x^{2}} - \frac{54}{x^{3}}$

$f(x) = (x^{-1} + x^{-2})(3x^{3}+27)$ $\Rightarrow f'(x) = (x^{-1} + x^{-2})(\frac{d}{dx}[3x^{3}+27]) + (3x^{3}+27)(\frac{d}{dx}[x^{-1} + x^{-2}])$ $= (x^{-1} + x^{-2})(9x^{2}) + (3x^{3}+27)(-x^{-2} -2x^{-3}) $ $= (9x + 9) + (-3x -27x^{-2} - 6 -54x^{-3})$ $= 6x + 3 - \frac{27}{x^{2}} - \frac{54}{x^{3}}$