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

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