Find the difference quotient of $LaTeX: \displaystyle f(x)=- 5 x^{2} + 9 x + 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=9 h + 9 x - 5 \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 5 h^{2} - 10 h x + 9 h - 5 x^{2} + 9 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 5 h^{2} - 10 h x + 9 h - 5 x^{2} + 9 x + 9)-(- 5 x^{2} + 9 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 5 h^{2} - 10 h x + 9 h}{h}=- 5 h - 10 x + 9$