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 two are essentially the same formula stated in different ways. They can be derived from each other as follows:
Note that $$\frac{d}{dx}(e^x)=e^x \ln(e) = e^x$$ is a special case of the formula for $a^x$ because $e$ has the special property that $\ln (e) =1$
Also $a^x=e^{\ln(a) x}$, which is another way into the derivative for $a^x$.
$$\frac{d}{dx}(e^{rx})=re^{rx}$$ by the chain rule. Let $r=\ln (a)$.
