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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- h - x + 9 \left(h + x\right)^{3} + 3 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=9 h^{3} + 27 h^{2} x + 3 h^{2} + 27 h x^{2} + 6 h x - h + 9 x^{3} + 3 x^{2} - x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(9 h^{3} + 27 h^{2} x + 3 h^{2} + 27 h x^{2} + 6 h x - h + 9 x^{3} + 3 x^{2} - x - 8)-(9 x^{3} + 3 x^{2} - x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{9 h^{3} + 27 h^{2} x + 3 h^{2} + 27 h x^{2} + 6 h x - h}{h}=9 h^{2} + 27 h x + 3 h + 27 x^{2} + 6 x - 1$