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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=h + x - 9 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 9 h^{2} - 18 h x + h - 9 x^{2} + x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 9 h^{2} - 18 h x + h - 9 x^{2} + x - 7)-(- 9 x^{2} + x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 9 h^{2} - 18 h x + h}{h}=- 9 h - 18 x + 1$