Find the difference quotient of $LaTeX: \displaystyle f(x)=4 x^{2} - 2 x - 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 2 h - 2 x + 4 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=4 h^{2} + 8 h x - 2 h + 4 x^{2} - 2 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(4 h^{2} + 8 h x - 2 h + 4 x^{2} - 2 x - 7)-(4 x^{2} - 2 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{4 h^{2} + 8 h x - 2 h}{h}=4 h + 8 x - 2$