Find the difference quotient of $LaTeX: \displaystyle f(x)=- 6 x^{2} - 9 x - 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x - 6 \left(h + x\right)^{2} - 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 6 h^{2} - 12 h x - 9 h - 6 x^{2} - 9 x - 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 6 h^{2} - 12 h x - 9 h - 6 x^{2} - 9 x - 1)-(- 6 x^{2} - 9 x - 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 6 h^{2} - 12 h x - 9 h}{h}=- 6 h - 12 x - 9$