1. **State the problem:** We want to find the derivative of the function $f(x) = a^x$, where $a$ is a positive constant. 2. **Recall the formula for the derivative of an exponential function:** The derivative of $a^x$ can be found using the natural exponential function and logarithms. We use the fact that $a^x = e^{x \ln(a)}$. 3. **Apply the chain rule:** $$\frac{d}{dx} a^x = \frac{d}{dx} e^{x \ln(a)} = e^{x \ln(a)} \cdot \frac{d}{dx} (x \ln(a))$$ 4. **Differentiate the exponent:** Since $\ln(a)$ is a constant, $$\frac{d}{dx} (x \ln(a)) = \ln(a)$$ 5. **Combine the results:** $$\frac{d}{dx} a^x = e^{x \ln(a)} \cdot \ln(a) = a^x \ln(a)$$ 6. **Final answer:** $$\boxed{\frac{d}{dx} a^x = a^x \ln(a)}$$ This means the derivative of $a^x$ is the original function multiplied by the natural logarithm of the base $a$.