# Thread: Deriving the number e

1. ## Deriving the number e

Is it possible to derive the number e, starting with the assumption that there exists a base number for an exponential function that is equal to its own derivative?

I've seen proofs that [e^x]' = e^x, using the definition of e. What I'm looking for is a way to start with the assumption that the above is true, and get to the limit definition of e.

I searched around a bit, without any luck, so I thought I'd ask here.

2. ## Re: Deriving the number e

Originally Posted by gralla55
Is it possible to derive the number e, starting with the assumption that there exists a base number for an exponential function that is equal to its own derivative? What I'm looking for is a way to start with the assumption that the above is true, and get to the limit definition of e.
If not, please try to explain exactly what you mean.

3. ## Re: Deriving the number e

I've attached a picture which should clarfy. What I mean is if there is a way to start with the assumption that there exisits a number^x which is its own derivative, and solve for that number to get to the last expression in my picture.

4. ## Re: Deriving the number e

So you're looking for a proof that:
$\lim_{x \to 0} (1+x)^{\frac{1}{x}}=e$
across the derivative of $e^x=e^x$
...?

5. ## Re: Deriving the number e

Yes.

I've seen proofs that [e^x]' = e^x, using the definition of e. What I'm looking for is how to do it the other way around. Starting with [e^x]' = e^x, and getting to the definition of e.

6. ## Re: Deriving the number e

Originally Posted by gralla55
I've attached a picture which should clarfy. What I mean is if there is a way to start with the assumption that there exisits a number^x which is its own derivative, and solve for that number to get to the last expression in my picture.

$\lim_{ x \rightarrow 0} (1+x)^{\frac{1}{x}}$ (1)

... then the most confortable way is to use the identity...

$(1+x)^{\frac{1}{x}}= e^{\frac{\ln (1+x)}{x}}$ (2)

... and compute the limit of the exponent in (2) using series expansion or l'Hopital rule...

Kind regards

$\chi$ $\sigma$