Find the difference quotient of $LaTeX: \displaystyle f(x)=- 10 x^{3} - 10 x^{2} - x - 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- h - x - 10 \left(h + x\right)^{3} - 10 \left(h + x\right)^{2} - 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 10 h^{3} - 30 h^{2} x - 10 h^{2} - 30 h x^{2} - 20 h x - h - 10 x^{3} - 10 x^{2} - x - 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 10 h^{3} - 30 h^{2} x - 10 h^{2} - 30 h x^{2} - 20 h x - h - 10 x^{3} - 10 x^{2} - x - 3)-(- 10 x^{3} - 10 x^{2} - x - 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 10 h^{3} - 30 h^{2} x - 10 h^{2} - 30 h x^{2} - 20 h x - h}{h}=- 10 h^{2} - 30 h x - 10 h - 30 x^{2} - 20 x - 1$