Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{2} + 9 x - 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=9 h + 9 x + 5 \left(h + x\right)^{2} - 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{2} + 10 h x + 9 h + 5 x^{2} + 9 x - 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{2} + 10 h x + 9 h + 5 x^{2} + 9 x - 1)-(5 x^{2} + 9 x - 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{2} + 10 h x + 9 h}{h}=5 h + 10 x + 9$