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 letter $e$ denotes this  number, whereas the letter $a$ can be any positive real number.
Because $\ln(e)=1$ (essentially by definition), the rule
$$\frac{d}{dx}(e^x)=e^x$$
is consistent with, and indeed a special case of, the rule
$$\frac{d}{dx}(a^x)=a^x\ln(a)$$
There is no "choosing when to use one or use the other".
