Boundedness of differential operator

Question

I wanna see if differential operator $D=\frac{d}{dx}$ is bounded or unbounded on $L^2[0,1]$ with $L^2-$norm(I know it's unbounded with sup norm). anyway, I don't have any ideas for proof, so any help or hints on at least boundedness of this operator is very appreciated.

Answer 1

Take $x_n (t) =t^n $ for $t\in [0,1]$ then $$||x_n ||_{L^2}=\frac{1}{\sqrt{2n+1}}\leq 1$$ $Dx_n \in L^2 [0,1]$ and $$||Dx_n ||_{L^2 } =\sqrt{\int_0^1 (nt^{n-1} )^2 dt}=\frac{n}{\sqrt{2n-1}}\to\infty$$ therefore the $D$ operator is unbounded.