What is the difference between exponential symbol $a^x$ and $e^x$ in mathematics symbols? I want to know the difference between the exponential symbol $a^x$ and $e^x$ in mathematics symbols and please give me some examples for both of them.
I asked this question because of the derivative rules table below contain both exponential symbol $a^x$ and $e^x$ and I don't know when should I use one of them and when should I use the another one.
Derivative rules table:

[Derivative rules table source]

The derivative of $e^x$ is $e^x\ln(e)$, so it's not different in that respect from other bases.  But $\ln(e)$ is $1$.
That's what's "natural" about the number $e$.
