1. Problem: Use the definition of the derivative to find $f'(x)$ for $f(x) = a^x$. 2. The definition of the derivative is: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. Applying this to $f(x) = a^x$: $$f'(x) = \lim_{h \to 0} \frac{a^{x+h} - a^x}{h} = \lim_{h \to 0} \frac{a^x a^h - a^x}{h} = a^x \lim_{h \to 0} \frac{a^h - 1}{h}$$ 4. The limit $\lim_{h \to 0} \frac{a^h - 1}{h}$ is the definition of the derivative of $a^x$ at 0, which equals $\ln(a)$. 5. Therefore: $$f'(x) = a^x \ln(a)$$ Final answer: $f'(x) = a^x \ln(a)$