Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{3} - 10 x^{2} + 10 x - 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=10 h + 10 x + 5 \left(h + x\right)^{3} - 10 \left(h + x\right)^{2} - 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{3} + 15 h^{2} x - 10 h^{2} + 15 h x^{2} - 20 h x + 10 h + 5 x^{3} - 10 x^{2} + 10 x - 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{3} + 15 h^{2} x - 10 h^{2} + 15 h x^{2} - 20 h x + 10 h + 5 x^{3} - 10 x^{2} + 10 x - 1)-(5 x^{3} - 10 x^{2} + 10 x - 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{3} + 15 h^{2} x - 10 h^{2} + 15 h x^{2} - 20 h x + 10 h}{h}=5 h^{2} + 15 h x - 10 h + 15 x^{2} - 20 x + 10$