Find the difference quotient of $LaTeX: \displaystyle f(x)=10 x^{2} + 5 x + 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=5 h + 5 x + 10 \left(h + x\right)^{2} + 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=10 h^{2} + 20 h x + 5 h + 10 x^{2} + 5 x + 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(10 h^{2} + 20 h x + 5 h + 10 x^{2} + 5 x + 1)-(10 x^{2} + 5 x + 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{10 h^{2} + 20 h x + 5 h}{h}=10 h + 20 x + 5$