Find the difference quotient of $LaTeX: \displaystyle f(x)=9 x^{2} + 7 x + 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=7 h + 7 x + 9 \left(h + x\right)^{2} + 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=9 h^{2} + 18 h x + 7 h + 9 x^{2} + 7 x + 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(9 h^{2} + 18 h x + 7 h + 9 x^{2} + 7 x + 3)-(9 x^{2} + 7 x + 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{9 h^{2} + 18 h x + 7 h}{h}=9 h + 18 x + 7$