Find the difference quotient of $LaTeX: \displaystyle f(x)=10 x^{2} + 2 x - 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=2 h + 2 x + 10 \left(h + x\right)^{2} - 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=10 h^{2} + 20 h x + 2 h + 10 x^{2} + 2 x - 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(10 h^{2} + 20 h x + 2 h + 10 x^{2} + 2 x - 9)-(10 x^{2} + 2 x - 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{10 h^{2} + 20 h x + 2 h}{h}=10 h + 20 x + 2$