derivatives
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:
Differentiability is a stronger condition than continuity. If $f$ is differentiable at $x=a$, then $f$ is continuous at $x=a$ as well. But the reverse need not hold.
Continuity of $f$ at $x=a$ requires only that $f(x)-f(a)$ converges to zero as $x\rightarrow a$.
For differentiability, that difference is required to converge even after being divided by $x-a$. In other words, $\dfrac{f(x)-f(a)}{x-a}$ must converge as $x\rightarrow a$.
Not that if that fraction does converge, the numerator necessarily converges to zero, implying continuity as I mentioned in the first paragraph.
For a good geometric understanding of the difference between $\Delta r$ and $dr$, look at how a Riemann Summation becomes a Riemann Integral
The width of each column starts as a $\Delta r$ but shrinks to a $dr$ the closer the point spacing.
Best Answer
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".