Find the difference quotient of $LaTeX: \displaystyle f(x)=6 x^{3} - 7 x^{2} + 3 x + 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=3 h + 3 x + 6 \left(h + x\right)^{3} - 7 \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=6 h^{3} + 18 h^{2} x - 7 h^{2} + 18 h x^{2} - 14 h x + 3 h + 6 x^{3} - 7 x^{2} + 3 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(6 h^{3} + 18 h^{2} x - 7 h^{2} + 18 h x^{2} - 14 h x + 3 h + 6 x^{3} - 7 x^{2} + 3 x + 9)-(6 x^{3} - 7 x^{2} + 3 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{6 h^{3} + 18 h^{2} x - 7 h^{2} + 18 h x^{2} - 14 h x + 3 h}{h}=6 h^{2} + 18 h x - 7 h + 18 x^{2} - 14 x + 3$