Why do all elementary functions have an elementary derivative?

Just think of how we find those elementary functions:

• We start with the constant functions, which have derivative $0$, and the identity function $f(x)=x$ which has derivative $1$.

• We combine functions by means of addition, subtraction, multiplication, division, composition. For all of those cases we have explicit rules for the derivative.

• We define new functions as the integral of other functions (e.g. $\ln x$ as integral of $1/x$). Obviously when deriving those we get back the function we started with.

• We define functions as the inverse of another function. Again, we've got an explicit formula for derivatives of inverse functions.

Any function that cannot be defined by a chain of such operations (and also some which can, using the integration rule) we don't consider elementary.

So basically the reason is in the way we construct elementary functions. In some sense, one could say it is because of what functions we consider elementary.

Indeed, this hold not only for elementary functions; even most non-elementary functions we use are defined through such operations (in particular by integrals).

The short answer is that we have differentiation rules for all the elementary functions, and we have differentiation rules for every way we can combine elementary functions (addition, multiplication, composition), where the derivative of a combination of two functions may be expressed using the functions, their derivatives and the different forms of combination.

Integration, on the other hand, neither has a direct rule for multiplication of two functions nor for composition of two functions. We can integrate the corresponding rules for differentiation and get something that looks like it (integration by parts and substitution), but it only works if you're lucky with what elementary functions are combined in what way.

You might say that there is a hope that there are rules out there, that we just haven't found them yet. This is not true; it's been proven that there are always integrals of elementary functions that are not elementary themselves (under most reasonable definitions of "elementary functions"). It's a deep result known as Liouville's theorem.

$\exp$, $\sin$, $\cos$, $\log$... are solutions of very simple differential equations. This fact plus the existence of explicit rules for the derivatives of sum/difference/product/quotient/composition guarantee that no "new" functions will appear when deriving elementary functions.