TITLE: Simple proof of irrationality of e
QUESTION [5 upvotes]: Is this reasoning correct?
Assume $e=\frac{p}{q}$ where $p$ and $q$ are natural numbers
taking natural log on both sides
and using the fact that natural $\ln e = 1$. We come up with 
$1 = \ln\left(\frac{p}{q}\right)$.
Taking derivative on both sides
$0 = \large\frac{1}{\frac{p}{q}}$
taking $\frac{p}{q}$ to other side
results in  $0 = 1$ absurdity 
therefore $e=\frac{p}{q}$ is false
and hence $e$ is irrational.
What do you think guys?

REPLY [6 votes]: Before differentiating, you actually come up with $1 = \ln e$. So what are you taking the derivative with respect to? $e$? I'm sure you know that differentiating with respect to a constant doesn't make sense. ;-)