Seems pretty reasonable to me. If you look at these questions and learn how to do them, you'll have learned a fair bit. It's not like you can learn by rote all the transformations you'll need to answer these, you're better off just learning the theory.
I would have loved to have a simple 100 known-but-non-trivial questions like this. You avoid the lottery of having to remember some particular detail (say some integral that appears in some derivation that an adhoc question might contain), but you don't avoid having to actually know how the theory works, because it's a little bit too hard to memorize.
Admittedly you might still get stuck on a trivial step but at least you've had a chance to go over the questions, and it might be relatively fresh.
I think these questions are meant to really address mechanical skills. The second question took me 7 differentiations. It takes understanding to be able to do that but the rate of error is going to be very high even for those that understand differentiation perfectly.
Edit: someone posted a link showing that one can generalize this result for g, f and their inverses. Interestingly, after reading up on it, the point Arnold is trying to make is that modern students have a poor grasp of infinitesimals.
I would have loved to have a simple 100 known-but-non-trivial questions like this. You avoid the lottery of having to remember some particular detail (say some integral that appears in some derivation that an adhoc question might contain), but you don't avoid having to actually know how the theory works, because it's a little bit too hard to memorize.
Admittedly you might still get stuck on a trivial step but at least you've had a chance to go over the questions, and it might be relatively fresh.