Let’s compute the derivative of the exponential function \( f(x) = a^x \) as the limit of the difference quotient:

\[ f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \]

Applying this definition to the function \( f(x) = a^x \), we obtain:

\[ f'(x_0) = \lim_{x \to x_0} \frac{a^x - a^{x_0}}{x - x_0} \]

We rewrite \( a^x = a^{x_0} \cdot a^{x-x_0} \), so:

\[ f'(x_0) = a^{x_0} \lim_{x \to x_0} \frac{a^{x-x_0} - 1}{x - x_0} \]

Introducing an auxiliary variable \( u = x - x_0 \) to simplify the calculation:

\[ L = \lim_{u \to 0} \frac{a^u - 1}{u} \qquad (\text{Notable Limit})\]

The value of \( L \) is the natural logarithm of the base \( a \), i.e., \( \ln(a) \). Therefore, the derivative of the exponential function is:

\[ f'(x) = a^x \cdot \ln(a) \quad , \quad \forall x \in \mathbb{R} \]