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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=7 h + 7 x + 6 \left(h + x\right)^{2} - 6$ and expanding gives $LaTeX: \displaystyle f(x+h)=6 h^{2} + 12 h x + 7 h + 6 x^{2} + 7 x - 6$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(6 h^{2} + 12 h x + 7 h + 6 x^{2} + 7 x - 6)-(6 x^{2} + 7 x - 6)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{6 h^{2} + 12 h x + 7 h}{h}=6 h + 12 x + 7$