Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{3} - 6 x^{2} - x - 6$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- h - x + 5 \left(h + x\right)^{3} - 6 \left(h + x\right)^{2} - 6$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{3} + 15 h^{2} x - 6 h^{2} + 15 h x^{2} - 12 h x - h + 5 x^{3} - 6 x^{2} - x - 6$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{3} + 15 h^{2} x - 6 h^{2} + 15 h x^{2} - 12 h x - h + 5 x^{3} - 6 x^{2} - x - 6)-(5 x^{3} - 6 x^{2} - x - 6)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{3} + 15 h^{2} x - 6 h^{2} + 15 h x^{2} - 12 h x - h}{h}=5 h^{2} + 15 h x - 6 h + 15 x^{2} - 12 x - 1$