Find the difference quotient of $LaTeX: \displaystyle f(x)=9 x^{2} - 2 x - 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 2 h - 2 x + 9 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=9 h^{2} + 18 h x - 2 h + 9 x^{2} - 2 x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(9 h^{2} + 18 h x - 2 h + 9 x^{2} - 2 x - 8)-(9 x^{2} - 2 x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{9 h^{2} + 18 h x - 2 h}{h}=9 h + 18 x - 2$