Find the difference quotient of $LaTeX: \displaystyle f(x)=3 x^{3} - 9 x^{2} - 4 x - 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 4 h - 4 x + 3 \left(h + x\right)^{3} - 9 \left(h + x\right)^{2} - 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=3 h^{3} + 9 h^{2} x - 9 h^{2} + 9 h x^{2} - 18 h x - 4 h + 3 x^{3} - 9 x^{2} - 4 x - 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(3 h^{3} + 9 h^{2} x - 9 h^{2} + 9 h x^{2} - 18 h x - 4 h + 3 x^{3} - 9 x^{2} - 4 x - 5)-(3 x^{3} - 9 x^{2} - 4 x - 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{3 h^{3} + 9 h^{2} x - 9 h^{2} + 9 h x^{2} - 18 h x - 4 h}{h}=3 h^{2} + 9 h x - 9 h + 9 x^{2} - 18 x - 4$