Find the difference quotient of $LaTeX: \displaystyle f(x)=x^{3} - 7 x^{2} + 8 x - 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=8 h + 8 x + \left(h + x\right)^{3} - 7 \left(h + x\right)^{2} - 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=h^{3} + 3 h^{2} x - 7 h^{2} + 3 h x^{2} - 14 h x + 8 h + x^{3} - 7 x^{2} + 8 x - 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(h^{3} + 3 h^{2} x - 7 h^{2} + 3 h x^{2} - 14 h x + 8 h + x^{3} - 7 x^{2} + 8 x - 5)-(x^{3} - 7 x^{2} + 8 x - 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{h^{3} + 3 h^{2} x - 7 h^{2} + 3 h x^{2} - 14 h x + 8 h}{h}=h^{2} + 3 h x - 7 h + 3 x^{2} - 14 x + 8$