Derivation/Linear/Example

Let ${}C^{0}(\mathbb {R} ,\mathbb {R} )$ denote the space of continuous functions from ${}\mathbb {R}$ to ${}\mathbb {R}$, and let ${}C^{1}(\mathbb {R} ,\mathbb {R} )$ denote the space of continuously differentiable functions. Then the mapping

$D\colon C^{1}(\mathbb {R} ,\mathbb {R} )\longrightarrow C^{0}(\mathbb {R} ,\mathbb {R} ),f\longmapsto f',$

which assigns to a function its derivative, is linear. In analysis, we prove that

${}(af+bg)'=af'+bg'\,$

holds for ${}a,b\in \mathbb {R}$ and another function ${}g\in C^{1}(\mathbb {R} ,\mathbb {R} )$.