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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=h + x - 7 \left(h + x\right)^{2} + 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 7 h^{2} - 14 h x + h - 7 x^{2} + x + 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 7 h^{2} - 14 h x + h - 7 x^{2} + x + 7)-(- 7 x^{2} + x + 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 7 h^{2} - 14 h x + h}{h}=- 7 h - 14 x + 1$