Find the difference quotient of $LaTeX: \displaystyle f(x)=6 x^{2} - 5 x + 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 5 h - 5 x + 6 \left(h + x\right)^{2} + 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=6 h^{2} + 12 h x - 5 h + 6 x^{2} - 5 x + 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(6 h^{2} + 12 h x - 5 h + 6 x^{2} - 5 x + 1)-(6 x^{2} - 5 x + 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{6 h^{2} + 12 h x - 5 h}{h}=6 h + 12 x - 5$