Find the difference quotient of $LaTeX: \displaystyle f(x)=7 x^{3} - 5 x^{2} + 6 x - 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=6 h + 6 x + 7 \left(h + x\right)^{3} - 5 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=7 h^{3} + 21 h^{2} x - 5 h^{2} + 21 h x^{2} - 10 h x + 6 h + 7 x^{3} - 5 x^{2} + 6 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(7 h^{3} + 21 h^{2} x - 5 h^{2} + 21 h x^{2} - 10 h x + 6 h + 7 x^{3} - 5 x^{2} + 6 x - 7)-(7 x^{3} - 5 x^{2} + 6 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{7 h^{3} + 21 h^{2} x - 5 h^{2} + 21 h x^{2} - 10 h x + 6 h}{h}=7 h^{2} + 21 h x - 5 h + 21 x^{2} - 10 x + 6$