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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x - 7 \left(h + x\right)^{3} + 3 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 7 h^{3} - 21 h^{2} x + 3 h^{2} - 21 h x^{2} + 6 h x - 9 h - 7 x^{3} + 3 x^{2} - 9 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 7 h^{3} - 21 h^{2} x + 3 h^{2} - 21 h x^{2} + 6 h x - 9 h - 7 x^{3} + 3 x^{2} - 9 x - 7)-(- 7 x^{3} + 3 x^{2} - 9 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 7 h^{3} - 21 h^{2} x + 3 h^{2} - 21 h x^{2} + 6 h x - 9 h}{h}=- 7 h^{2} - 21 h x + 3 h - 21 x^{2} + 6 x - 9$