Find the difference quotient of $LaTeX: \displaystyle f(x)=8 x^{2} - x - 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- h - x + 8 \left(h + x\right)^{2} - 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=8 h^{2} + 16 h x - h + 8 x^{2} - x - 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(8 h^{2} + 16 h x - h + 8 x^{2} - x - 3)-(8 x^{2} - x - 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{8 h^{2} + 16 h x - h}{h}=8 h + 16 x - 1$