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

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