1. **State the problem:** We are given the function $f(x) = \frac{13}{x}$ and need to simplify the difference quotient: $$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$$ 2. **Write the expressions for $f(x+h)$ and $f(x)$:** $$f(x+h) = \frac{13}{x+h}$$ $$f(x) = \frac{13}{x}$$ 3. **Substitute into the difference quotient:** $$\frac{\frac{13}{x+h} - \frac{13}{x}}{h}$$ 4. **Find a common denominator for the numerator:** $$\frac{13x - 13(x+h)}{(x+h)x} = \frac{13x - 13x - 13h}{(x+h)x} = \frac{-13h}{(x+h)x}$$ 5. **Rewrite the difference quotient:** $$\frac{\frac{-13h}{(x+h)x}}{h} = \frac{-13h}{(x+h)x} \times \frac{1}{h}$$ 6. **Cancel $h$ in numerator and denominator:** $$\frac{-13\cancel{h}}{(x+h)x} \times \frac{1}{\cancel{h}} = \frac{-13}{(x+h)x}$$ 7. **Final simplified form:** $$\boxed{\frac{f(x+h) - f(x)}{h} = \frac{-13}{(x+h)x}}$$ This expression represents the average rate of change of the function $f(x)$ over the interval from $x$ to $x+h$.