-1

I watched an MIT video about the Euler number. There they figure it out as follows:

The exponential function should be a function that per definition has the property, that it equals to its derivative.

So if $x=0$

$e^{x} = 1$, so the derivative 1 too. But then $e^{x}$ must be $1 + x$, but then the derivative too, but then $e^{x}$ must be $1 + x + \frac{1}{2}x^{2}$ and so on.

Why can we keep the sum of the previous results?

So why wouldn't be ok, if we would just say:

$e^{x}$ = 1, then the derivative is 1 too, but then $e^{x}$ must be !! x !! simply. So we wouldn't start the series.

I hope I am clear enough.

Here is the part of the video, it takes only 1 minute, so you can see better what I mean. At 7:11

user3435407
  • 1,099

1 Answers1

0

At the start it says that it is assumed that at $x=0$ the value is $1$, so you can not take $x$ as it is not $1$ at $x=0$.

quid
  • 42,135
  • yes, x=0, so that's why $e^{0} = 1$. So it's derivative should be 1 too. But if its derivative is 1, then the original function must be 1 + x. But why not just x? – user3435407 Jul 11 '15 at 19:55
  • Because then it is not $1$ at $0$ anymore. But also because then its derivative is $1$ which is not $x$, so you need to put the $1$ back there. – quid Jul 11 '15 at 20:06