Find the difference quotient of $LaTeX: \displaystyle f(x)=- 9 x^{3} + 7 x^{2} - 8 x + 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 8 h - 8 x - 9 \left(h + x\right)^{3} + 7 \left(h + x\right)^{2} + 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 9 h^{3} - 27 h^{2} x + 7 h^{2} - 27 h x^{2} + 14 h x - 8 h - 9 x^{3} + 7 x^{2} - 8 x + 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 9 h^{3} - 27 h^{2} x + 7 h^{2} - 27 h x^{2} + 14 h x - 8 h - 9 x^{3} + 7 x^{2} - 8 x + 7)-(- 9 x^{3} + 7 x^{2} - 8 x + 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 9 h^{3} - 27 h^{2} x + 7 h^{2} - 27 h x^{2} + 14 h x - 8 h}{h}=- 9 h^{2} - 27 h x + 7 h - 27 x^{2} + 14 x - 8$