I am on derivatives at the moment and I just bumped into this number $e$, "Euler's number" . I am told that this number is special especially when I take the derivative of $e^x$ , because its slope of any point is 1. Also it is an irrational ($2.71828\ldots$) number that never ends, like $\pi$.
So I have two questions, I can't understand
- What is so special about this fact that it's slope is always 1?
- Where do we humans use this number that is so useful, how did Mr Euler come up with this number?
and how come this number is a constant? where can we find this number in nature?
Answer
You don't take the derivative of a constant. You could, but it's zero.
What you should be talking about is the exponential function, $ e^x $ commonly denoted by $ \exp(\cdot ) $. Its derivative at any point is equal to its value, i.e. $ \frac{d}{dx} e^x \mid_{x = a} = e^a $. That is to say, the slope of the function is equal to its value for all values of $ x $.
As for how to arrive at it, it depends entirely on definition. There are numerous ways to define $ e $, the exponential function, or the natural logarithm. One common definition is to define $$ \ln x := \int\limits_1^x \frac{1}{t} \ dt $$ From here, you can define $ e $ as the sole positive real such that $ \ln x = 1 $ and arrive at it numerically.
Another common definition is $ e = \lim\limits_{n \to \infty}\left(1 + \frac{1}{n}\right)^n $, although in my opinion it is easier to derive properties from the former definition.
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