Find the difference quotient of $LaTeX: \displaystyle f(x)=7 x^{3} + 9 x^{2} + x + 2$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=h + x + 7 \left(h + x\right)^{3} + 9 \left(h + x\right)^{2} + 2$ and expanding gives $LaTeX: \displaystyle f(x+h)=7 h^{3} + 21 h^{2} x + 9 h^{2} + 21 h x^{2} + 18 h x + h + 7 x^{3} + 9 x^{2} + x + 2$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(7 h^{3} + 21 h^{2} x + 9 h^{2} + 21 h x^{2} + 18 h x + h + 7 x^{3} + 9 x^{2} + x + 2)-(7 x^{3} + 9 x^{2} + x + 2)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{7 h^{3} + 21 h^{2} x + 9 h^{2} + 21 h x^{2} + 18 h x + h}{h}=7 h^{2} + 21 h x + 9 h + 21 x^{2} + 18 x + 1$