Find the difference quotient of $LaTeX: \displaystyle f(x)=x^{2} - 6 x - 10$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 6 h - 6 x + \left(h + x\right)^{2} - 10$ and expanding gives $LaTeX: \displaystyle f(x+h)=h^{2} + 2 h x - 6 h + x^{2} - 6 x - 10$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(h^{2} + 2 h x - 6 h + x^{2} - 6 x - 10)-(x^{2} - 6 x - 10)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{h^{2} + 2 h x - 6 h}{h}=h + 2 x - 6$