Find the difference quotient of $LaTeX: \displaystyle f(x)=8 x^{3} - x^{2} + 9 x + 10$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=9 h + 9 x + 8 \left(h + x\right)^{3} - \left(h + x\right)^{2} + 10$ and expanding gives $LaTeX: \displaystyle f(x+h)=8 h^{3} + 24 h^{2} x - h^{2} + 24 h x^{2} - 2 h x + 9 h + 8 x^{3} - x^{2} + 9 x + 10$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(8 h^{3} + 24 h^{2} x - h^{2} + 24 h x^{2} - 2 h x + 9 h + 8 x^{3} - x^{2} + 9 x + 10)-(8 x^{3} - x^{2} + 9 x + 10)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{8 h^{3} + 24 h^{2} x - h^{2} + 24 h x^{2} - 2 h x + 9 h}{h}=8 h^{2} + 24 h x - h + 24 x^{2} - 2 x + 9$