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

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