Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{2} - 10 x - 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 10 h - 10 x + 5 \left(h + x\right)^{2} - 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{2} + 10 h x - 10 h + 5 x^{2} - 10 x - 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{2} + 10 h x - 10 h + 5 x^{2} - 10 x - 3)-(5 x^{2} - 10 x - 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{2} + 10 h x - 10 h}{h}=5 h + 10 x - 10$