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]
 A: You should think of the exponential function as being the primitive object. It is the inverse to the logarithm $\log:(0,\infty)\rightarrow\mathbf{R}$ (when I write $\log$ I mean the natural log, also denoted sometimes as $\ln$, not the base 10 logarithm). That is, $\log(e^x))=x$ and $e^{\log(y)}=y$ for $x\in\mathbf{R}$ and $y\in(0,\infty)$. 
If you know what the exponential function is, then you can understand the function given by sending $x$ to $a^x$ for a positive real number $a$. By definition, $a^x$ means $e^{x\log(a)}$. Remember that here $\log(a)$ is just a number, the (natural) logarithm of $a$. Keeping this in mind, and remembering that the derivative of the exponential function is exist, you can use the chain rule to derive
$(d/dx)(a^x)=(d/dx)(e^{x\log(a)})=e^{x\log(a)}(d/dx)(x\log(a))=\log(a)e^{x\log(a)}=\log(a)a^x$.
A: 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 $r=\ln (a)$.
A: 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".
A: 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 $e$.
