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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=h + x - \left(h + x\right)^{2} - 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=- h^{2} - 2 h x + h - x^{2} + x - 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- h^{2} - 2 h x + h - x^{2} + x - 4)-(- x^{2} + x - 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- h^{2} - 2 h x + h}{h}=- h - 2 x + 1$