How e derived?

Answer 1

e is derived in several different ways.

One way is the infinite sum:

e = 1 + 1/1! + 1/2! + 1/3! + ...

Another is to note that the function 2^x has a gradient of approx 0.6931*2^x while 3^x has a gradient of 1.0986*3^x. Therefore by continuity (and the intermediate value theorem), there must be a value between 2 and 3 such that the gradient of the curve has the same value as the curve. This value is e.