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

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