= Differentiation operator
{title2=$Df=f'$}
= Derivative operator
{synonym}
The differentiation operator sends a <differentiable function> to its <derivative>; its domain and the <norms> must be specified. On $C^1([-1,1])$ with values in $C([-1,1])$, using the <supremum norm> on both, it is an unbounded <linear operator>: $f_n(x)=\sin(nx)$ satisfies $\|f_n\|_\infty\leq1$ and $\|Df_n\|_\infty=n$. Its <graph norm> is $\|f\|_\infty+\|f'\|_\infty$, which is <complete>. Indeed, the <kernel> consists of constants and is <complete>, while $Sg(x)=\int_0^xg(t)\,dt$ is a bounded <right inverse> by the <fundamental theorem of calculus>; apply <completeness of a graph norm from a bounded right inverse>.
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