TITLE: Functions with no closed-form derivative
QUESTION [6 upvotes]: There are many well-known functions (such as $x^x$) which have no closed-form integral. However, are there any elementary functions whose derivatives cannot be expressed in a  closed-form manner?

REPLY [14 votes]: If $f(x)$ is built out of the "usual" functions (rational functions, exponentials, logarithms, trig, inverse trig) using addition, subtraction, multiplication, division, or composition, then $f'(x)$ is also.
The proof is on induction on complexity.  For the base case, all of "easy" functions work:  If $f(x) = x$ or $e^x$ or $\ln x$ or $\sin x$ or $\operatorname{trig}(x)$ or $\operatorname{arctrig}(x)$ (where "trig" stands for one of the 6 trig functions), then $f'(x)$ is elementary.
Next, suppose $f(x)$, $f'(x)$, $g(x)$, and $g'(x)$ are all elementary.
Is $f(x) + g(x)$ also elementary?  Yes because $(f(x)+g(x))' = f'(x) + g'(x)$ is a sum of elementary functions, hence is elementary.  The same argument works for subtraction.
For multiplication and division, use the product/quotient rules.  For composition, use the chain rule.
Then, e.g., composition works as follow:  $f(g(x))' = f'(g(x))g'(x)$.  But $f'(g(x))$ is elementary because $f'$ and $g$ are and elementary things are closed under composition.  Further, $g'(x)$ is elementary by assumption.  So, $f'(g(x))g'(x)$ is a product of elementary functions, hence elementary.