Find the difference quotient of $LaTeX: \displaystyle f(x)=- 4 x^{2} - x - 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- h - x - 4 \left(h + x\right)^{2} - 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 4 h^{2} - 8 h x - h - 4 x^{2} - x - 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 4 h^{2} - 8 h x - h - 4 x^{2} - x - 1)-(- 4 x^{2} - x - 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 4 h^{2} - 8 h x - h}{h}=- 4 h - 8 x - 1$