Find the difference quotient of $LaTeX: \displaystyle f(x)=9 x^{3} - 8 x^{2} - 6 x + 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 6 h - 6 x + 9 \left(h + x\right)^{3} - 8 \left(h + x\right)^{2} + 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=9 h^{3} + 27 h^{2} x - 8 h^{2} + 27 h x^{2} - 16 h x - 6 h + 9 x^{3} - 8 x^{2} - 6 x + 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(9 h^{3} + 27 h^{2} x - 8 h^{2} + 27 h x^{2} - 16 h x - 6 h + 9 x^{3} - 8 x^{2} - 6 x + 5)-(9 x^{3} - 8 x^{2} - 6 x + 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{9 h^{3} + 27 h^{2} x - 8 h^{2} + 27 h x^{2} - 16 h x - 6 h}{h}=9 h^{2} + 27 h x - 8 h + 27 x^{2} - 16 x - 6$