Two questions about Euler's number $e$ 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?
 A: 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. 
A: Just a slight correction, as Jon Claus notes about the derivative of $e^x$:
what you may be remembering is that "$e$ is the unique real number such that the value of the derivative (slope of the tangent line) of the function $f(x) = e^x$ at the point $x = 0$ is equal to $1$.
See the Wikipedia article on Euler's number $e$ for more fascinating information:


*

*The number e is the unique positive real number such that
$$\frac{d}{dt}e^t = e^t.$$

*The number e is the unique positive real number such that
$$\frac{d}{dt} \log_e t = \frac{1}{t}.$$
The following three characterizations can be proven equivalent:

*The number e is the limit
$$e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n$$
Similarly:
$$e = \lim_{x\to 0} \left( 1 + x \right)^{1/x}$$

*The number e is the sum of the infinite series
$$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$
where $n!$ is the factorial of n.

*The number e is the unique positive real number such that
$$\int_1^e \frac{1}{t} \, dt = 1.$$


The number $e$ is of eminent importance in mathematics, alongside $0, 1, \pi, \;\text{and}\; i.$ All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity: $$e^{i\pi} + 1 = 0$$ Like the constant $π, e$ is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. 

