TITLE: Could you explain why $\frac{d}{dx} e^x = e^x$ "intuitively"?
QUESTION [49 upvotes]: As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself.
thanks.

REPLY [2 votes]: A one-liner: $e^x$ is an eigenvector of the operator $\frac{d}{dx}$
Similarly to the way a vector $x$ can be an eigen-vector of a matrix operator $A$.
By the definition of derivation and the eigenvalue equation $Ax=\lambda x$, recursively one gets a representation (one of the known formulas) of $e^x$
What is an eigen-vector of an operator? It is this vector (part of the space where the operator acts) which is left unchanged (except a re-scaling) from the action of the operator. In other words, the eigen-vector is aligned ("parallel") to the action of the operator, thus not changed in other way.
How does this relate to derivation and exponential function.  Even more intuitively, derivation decimates the function it acts upon, exponentialy (consider $(x^n)'=nx^{n-1}$), so a function that is unchanged by such exponential decimation, must be exponential itself (eigen-vector) 
(to maintain the terminology used above, the function should be exponential to all orders, i.e $e^x$).