Why do you get e^x when you differentiate e^x

It all comes down to the how e is defined. e=1+(1/1!)+(1/2!)+(1/3!) all the way up to (1/infinity). e^x is equal to 1 + (x/1!)+((x^2)/2!)+((x^3/3!)) and again continues on to ((x^infinite)/factorial of infinity) now if you difference rewrite this function you get 1 + x/1 + (x^2)/(2x1) + (x^3)/(3x2x1)) and so on. If you were to differentiated that function the power would drop down by one but also would the factorial of the denominator because when cancelling out the highest multiplier in the factorial, it decreases the factorial by 1. This creates a chain reaction which goes on forever (as infinity is a power) and eventually shows that there is no change to the function (that function being e^x). You can do the same in reverse to prove it for integration too!