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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=h + x + 6 \left(h + x\right)^{3} - \left(h + x\right)^{2} + 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=6 h^{3} + 18 h^{2} x - h^{2} + 18 h x^{2} - 2 h x + h + 6 x^{3} - x^{2} + x + 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(6 h^{3} + 18 h^{2} x - h^{2} + 18 h x^{2} - 2 h x + h + 6 x^{3} - x^{2} + x + 8)-(6 x^{3} - x^{2} + x + 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{6 h^{3} + 18 h^{2} x - h^{2} + 18 h x^{2} - 2 h x + h}{h}=6 h^{2} + 18 h x - h + 18 x^{2} - 2 x + 1$