I am studying linear KdV equation $y_t+y_x+y_{xxx}=0$, in order to apply operator semi-group method, we need to define an operator $A=\frac{d}{dx}+\frac{d^3}{dx^3}:D(A)\subset L^2(0,1)\rightarrow L^2(0,1)$, where $D(A)=\{f\in H^3(0,1):f(0)=f(1)=0,f'(0)=0\}$, $Af=f'+f'''$ for $f\in D(A)$( derivative is in the sense of distribution). I have proven $A$ is a densely defined closed operator, so we have graph norm $\|\cdot\|_{D(A)}$ on $D(A)$, i.e., $$\|f\|_{D(A)}:=\|f\|_{L^2}+\|f'+f'''\|_{L^2}.$$ such that $(D(A),\|\cdot\|_{D(A)})$ is a Hilbert space, if I define another operator $B:\mathbb{R}\rightarrow D(A)'$ by $(Bu,f)=uf'(1)$, i.e., $Bu=-u\delta'_1$.
The question is, how to prove $B$ is a bounded operator, i.e., there exists $C>0$ such that for any $f\in D(A)$ and $u\in \mathbb{R}$, we have $|(Bu,f)|\leq C|u|\cdot \|f\|_{D(A)} $.
It is enough to verify there exists $C>0$ such that $|f'(1)|\leq C(\|f\|_{L^2}+\|f'+f'''\|_{L^2})$ for $f\in D(A)$.
My attempt:
Since $f(0)=f(1)=0$ and $H^3(0,1)\subset C^{2,1/2}[0,1]$, there exists $a\in (0,1)$ such that $f'(a)=0$, combine with $f'(0)=0$, we have $b\in (0,1)$ such that $f''(b)=0$, so that $$f'(1)=\int_0^1\int_b^t(f'(s)+f'''(s))\,dsdt-\int_0^1 f(t)\,dt+f(b).$$ the first term and second term can be controlled by $\|f'+f'''\|_{L^2}$ and $\|f\|_{L^2}$ respectively, but how to controll $f(b)$? Any help will be apreciated.