Differentiation always easy? There are many examples of real functions admitting antiderivatives (since e.g. continuous), but where computing a concrete antiderivative is a seriously hard problem even if an elementary one exists.
What about differentiation? My experience is that the basic rules of calculus along with term-by-term differentiantion of power series make differentiation a just-do-it kind of problem for virtually all everyday kinds of functions. In fact, if we add the limit exchange trick for uniformly convergent sequence of derivatives, I cannot think of any examples, where finding a closed form for $f'$, given a closed form for $f$, is not a mechanical task.
So the question is: are there any examples of real functions $f$, such that
(1) $f$ is given in "nice" closed form $f(x)=\ldots$
(2) it is "relatively easy" to justify that $f$ is differentiable
(3) computing the derivative of $f$ is actually hard.
This isn't exactly a precise question, but there just might be a "know it when I see it" example.
 A: I think one example you might be interested in is the Cantor Ternary Function. Recall that the Cantor set is formed by taking the unit interval $[0,1]$, then throwing away the middle third, then throwing away the middle thirds of each resulting interval, and doing so infinitely many times. The resulting set is the Cantor set. 
You can read the exact definition of the function in the link but loosely speaking, it's obvious that the function is constant on the complement of the Cantor ternary set: the intervals you threw away. So clearly the function has zero derivative on these intervals. Less obvious is that the function is not differentiable at any point of the Cantor ternary set. On the other hand, the Cantor ternary set has measure zero, so in a precise sense the function is differentiable almost everywhere. It's also continuous, which is not obvious! So it's an example of an increasing, continuous function whose derivative is 0 almost everywhere! 
A: Given any function comprised of elementary compositions of base case elementary functions whose derivatives are known, i.e. starting with base case elementary functions and then making new functions through repeated application of addition, multiplication, and composition of functions, the standard rules of differentiation say that we can come up with an elementary formulation for the derivative provided derivatives are "known" for the base case functions. Thus, your only hope for an example is something like an exotic function whose definition we take as given, in terms of base case exotic functions that are acceptable, but whose derivative cannot be similarly expressed in terms of the collection of "accepted" base case exotic functions. But this seems a bit pedantic. 
