Of the mappings between vector spaces, the most well behaving are the bounded linear operators, and the derivative doesn't belong to these. But yes, it's linear.
Edit: Originally wrote f(x) = k·sin(k·x), but meant f(x) = k·sin(x/k).
Additionally, it only really makes sense to talk about bounded operators between topological vectors spaces (as you need to make sense of what it means to be bounded), of which the most commonly dealt with are Banach spaces.
Reading your comment, I wondered how can you define bounded sets in a topological vector space (where you don't have a norm). The definition is cute: a set X if bounded if any neighborhood of 0 can be inflated to include the whole of X.
(Also, what's the most natural norm on C^k? The sup norm? I haven't done any functional analysis in years)
|f| = ∑ᵢ₌₀ᵏ sup|f⁽ⁱ⁾|.
On open domains you can also use the topology that forces uniform convergence on all compact subsets. But this will only give you a metric space, no Banach space (but you’ll include unbouded functions). This is needed for studying Brownian motions with an unbounded time domain.
Regarding the other question: If you take Cᵏ⁻¹ as a the co-domain differentiation will be continuous. IIRC to get unbounded linear maps defined on the whole Banach space you need the axiom of choice, you won’t be able to write one down.
The problem with differential operators is that they are usually only defined on a dense subset of the domain, and there they are not bounded. E.g. in quantum mechanics the space of states is L² but all the interesting observables are differential operators. You can weasel out of this situation by defining them on a subset of “physical states” (e.g. smooth wave functions of rapid decay). But they aren’t continuous anymore (the spectrum is unbounded). But on Hilbert spaces everything mostly works out fine. Physicist usually ignore those technical problems and still don’t make mistakes.
First course in functional analysis.
Before that, people usually take linear algebra, all the calculus courses, some kind of theorem-proving course for calculus (either analysis or real analysis or really rigorous versions of the calculus courses) and measure theory, maybe also topology.