... a continuous differential map $\dfrac{d}{dx} : C^k(\mathbb R)\rightarrow C^{k-1}(\mathbb R)$ ...
I was wondering why a differential map could from $C^k(\mathbb R)$ to $C^{k-1}(\mathbb R)$, rather than from $C^{k-1}(\mathbb R)$ to $C^{k}(\mathbb R)$?