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

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