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According to this video, about 8 minutes in, when doing calculus, write "in terms of $e$" to make things easier. What, exactly, is meant by this? What are some examples of things done "in terms of $e$" to make the maths easier, and what were the alternatives to not doing it in terms of $e$?

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    Haven't watched the video, but if you have any exponent $a^{f(x)}$ you want to rewrite it as $e^{\ln a f(x)}$ before you do any calculus; it's just a little easier to deal with. That's probably the kind of thing he's talking about. – 2017-01-10

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The convenience of the function $e^{x}$ lies in two facts. (1) This function is its own derivative. This can be most easily seen by examining its Taylor series, say at $x = 0$: $$ e^{x} = \sum_{k = 0}^{\infty} {x^{k} \over k!}, $$ and differentiating term by term (all the necessary convergence is there). This allows the use of $e^{x}$ to differentiate other exponential functions, like $2^x$.

(2) If $i$ is the imaginary unit ($i^2 = -1$), then $e^{i \theta} = \cos(\theta) + i \sin(\theta)$ (here $\theta$ is in radians). Thus, the function also plays a crucial role in complex analysis, hence in many other areas of the mathematics.

I would recommend studying the first chapter of H. Cartan's "Elementary theory of analytic functions".

Hope this helps.

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    Give thanks for Euler's formula for complex exponential functions saving my life so many times. – 2017-01-10
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I think the video (I couldn't find the exact time or sentence) is referring to writing an exponential in terms of the number e. E has many useful properties, but one important one is that the derivative of $e^{kx}$ with respect to x is $k\cdot e^{kx}$. For example, $\frac{d}{dx}e^{2x} = 2\cdot e^{2x}$

Another useful property is that $x^y=e^{\ln \left(x\right)\cdot y}$, where $ln$ is the natural logarithm, or logarithm base e. Using that formula, you can re-write exponentials "in terms of e." For example, $2^x=e^{\ln \left(2\right)\cdot x}\approx e^{0.693x}$

Using both formulas can make calculus much easier. For example, if you wanted to find $\frac{d}{dx}3^x$, you could rewrite the function in terms of e to get $\frac{d}{dx}e^{\ln \left(3\right)\cdot x}$, then take the derivative and get $\ln \left(3\right)\cdot e^{\ln \left(3\right)\cdot x}$

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    :-D first post? – 2017-01-10
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    @Simple Art Yup. I've looked at this site before, but I only joined recently (basically today, although I joined another section a couple of days ago). – 2017-01-10
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    Then we're glad your here so that you may [support us as we walk to better places](https://despair.com/collections/demotivators/products/be-the-bridge). $\ddot\smile$ – 2017-01-10
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Solving the basic differential equation $$\frac{\mathrm d}{\mathrm d t} y(t) = y(t),\quad y(0)=1 $$ yields $y(t) = e^t.$ If you were to use another base, say $2$, then the solution would be slightly more awkward, namely $$y(t) = 2^{t/\ln(2)}.$$

The natural logarithm is even tightly linked to $e$, so we don't really away with using $e$. In some sense it turns out that $e$ is the natural constant of growth.

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    AFAIK, the natural logarithm is defined to be the inverse of the natural exponential. – 2017-01-10
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    That's my point; we don't really get away with using the natural exponential. It *is* possible to define the natural logarithm without any mentioning of $e$, of course, but everything gets easier when we use $e$. – 2017-01-10
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When integrating rational functions, the answer is often in terms of natural logarithms, log base e. And more so, the derivative of e^x is e^x, likewise, the integral of e^x is e^x.

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Have you ever heard that radians are the natural unit for trigonometry? It is, after all, why we teach it. $e$ is the same way, but for exponential functions. It is not necessarily something to do with calculus, though it is strongly related, since,

$$\frac d{dx}e^x=e^x$$

Which is quite a useful property. For the same reason radians are good, because they both come out so simple, it was like it was meant to be.

An interesting result highly important to mathematics is Euler's formula:

$$e^{ix}=\cos x+i\sin x$$

Which uses both $e$ and radians.