Exponential Derivatives
Results
Given $a \in \R $, define $f: \R \to \R $ by
\[ f(x) = a^x \quad \text{for all } x \in \R \]
Then $f$ is differentiable and its derivative is the function $f': \R \to \R $ defined by\[ f'(x) = \ln(a)a^x \quad \text{for all } x \in \R \]
This proposition encompasses the special case $f(x) = e^x$ then $f'(x) = e^x$.