Derivatives Involving e^x

What is e?

e is a real number and it has 3 definitions:

1.
2.
3.	e is the number such that

If you want to use the first definition to see what e exactly equals to, you need to plug ¥ into (1 + 1/x)^x, which is impossible. However, if you just want to see what e  approximately equals to, you can use your calculator and plug in a large number, say 1 million, and see how much (1 + 1/1,000,000)^1,000,000 yields. The calculator should give you 2.718280469. And that is a very good approximation of e since e = 2.7182871828459...

e is not a terminating decimal since the ellipsis (...) signifies the decimal never ends, nor it seems that e is a repeating decimal. Thus, like p, e is an irrational number. 

The second definition of e is in terms of a summation. A summation is a just sum of terms (usually infinitely many terms and so is this case). Each term is of the form 1/n! where n! is the product of first n positive integers, i.e., n! = n(n – 1)(n – 2)...(3)(2)(1). For example, 4! = 4(3)(2)(1) = 24, 3! = 3(2)(1) = 6, 2! = 2(1) = 2, 1! = 1, and last but not least 0! = 1 (despite 0 is not a positive integer). Therefore, according to the second definition, e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ... Again there is no way we can add up all the terms (since there are infinitely many), however, if you just add up the first 11 terms, i.e., 1/0! + 1/1! + 1/2! + 1/3! + ... + 1/10!, you will get 2.718281801 (using the calculator, of course)—an even better approximation! 

The third definition does not give an approximation of e, however, it’s very useful to find the derivative of f(x) = e^x.

Recall the definition of f´(x), derivative of f(x):

Hey, the derivative of e^x is itself!