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

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