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

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