Find the difference quotient of $LaTeX: \displaystyle f(x)=9 x^{2} + 2 x - 1$ .

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