Proving the Derivative of e^x Is e^x

Date: 02/20/2006 at 00:17:22
From: Ethan
Subject: Derivation of e^x is e^x

What is the delta x proof of (dy/dx)e^x = e^x?  I know using delta x 
you can prove (dx/dy) of x^3 = 3x^2 but I can't work out e^x.

Date: 02/20/2006 at 09:06:43
From: Doctor George
Subject: Re: Derivation of e^x is e^x

Hi Ethan,

Thanks for writing to Doctor Math.

Here is a proof that uses the definition of derivative, as you are
attempting to do.

  Proving That d(e^x)/dx = e^x
    http://mathforum.org/library/drmath/view/53731.html 

Here are two other approaches that may be of interest to you.

1. Apply the chain rule to ln(e^x) = x.

2. Look at the series expansion of e^x.  If you take the derivative of
each term in the series you will see that the result is just a
duplication of the series.

Does that make sense?  Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/