1. **State the problem:** Show that for the function $f(x) = 5^x$, the difference quotient is $$\frac{f(x+h) - f(x)}{h} = 5^x \frac{5^h - 1}{h}$$ 2. **Recall the function and difference quotient:** The difference quotient for a function $f$ is $$\frac{f(x+h) - f(x)}{h}$$ which measures the average rate of change of $f$ over the interval from $x$ to $x+h$. 3. **Evaluate $f(x+h)$:** Since $f(x) = 5^x$, then $$f(x+h) = 5^{x+h} = 5^x \cdot 5^h$$ using the exponent rule $a^{m+n} = a^m \cdot a^n$. 4. **Substitute into the difference quotient:** $$\frac{f(x+h) - f(x)}{h} = \frac{5^x \cdot 5^h - 5^x}{h}$$ 5. **Factor out $5^x$ from the numerator:** $$= \frac{5^x (5^h - 1)}{h}$$ 6. **Final expression:** Thus, $$\frac{f(x+h) - f(x)}{h} = 5^x \frac{5^h - 1}{h}$$ This completes the proof. **Explanation:** We used the property of exponents to rewrite $f(x+h)$ and factored $5^x$ out of the numerator to simplify the difference quotient expression.