Chapter 3 - Differentiation - 3.3 Product and Quotient Rules - Exercises - Page 122: 18

$-5x^{4}-\frac{1}{x^{2}}$

Using the product rule, we have $ f'(x)=\frac{d}{dx}(\frac{1}{x}-x^{2})\times(x^{3}+1)+(\frac{1}{x}-x^{2})\times\frac{d}{dx}(x^{3}+1)$ $=(-\frac{1}{x^{2}}-2x)(x^{3}+1)+(\frac{1}{x}-x^{2})3x^{2}$ $=(-x-\frac{1}{x^{2}}-2x^{4}-2x)+(3x-3x^{4})$ $=-5x^{4}-\frac{1}{x^{2}}$