Find the difference quotient of $LaTeX: \displaystyle f(x)=- 6 x^{3} + 8 x^{2} - x - 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- h - x - 6 \left(h + x\right)^{3} + 8 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 6 h^{3} - 18 h^{2} x + 8 h^{2} - 18 h x^{2} + 16 h x - h - 6 x^{3} + 8 x^{2} - x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 6 h^{3} - 18 h^{2} x + 8 h^{2} - 18 h x^{2} + 16 h x - h - 6 x^{3} + 8 x^{2} - x - 8)-(- 6 x^{3} + 8 x^{2} - x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 6 h^{3} - 18 h^{2} x + 8 h^{2} - 18 h x^{2} + 16 h x - h}{h}=- 6 h^{2} - 18 h x + 8 h - 18 x^{2} + 16 x - 1$