Let $u(x) \in H^1_0$ is a complex function in sobolev space extension by zero.

How to find $J'(u)$ for $$ J(u)= \int\limits_0^l |u(x)|^2 dx $$??

In $L_2$ it's eazy: $$ J'(u) = (\int\limits_0^l|u(x)|^2 dx)'=(||u(x)||^2)'= 2 u(x) $$ , but i didn't work with $H^1_0 (||u(x)||^2_{H^1_0} = \int\limits_0^l |u'(x)|^2 dx)$

Let $u(x) \in H^1_0$ is a complex function in sobolev space extension by zero.
How to find $J'(u)$ for $$ J(u)= \int\limits_0^l |u(x)|^2\operatorname{d\!}x\;?? $$ In $L_2$ it's easy: $$ J'(u) = \left(\int\limits_0^l|u(x)|^2 \operatorname{d\!}x\right)'=\big(\|u(x)\|^2\big)'= 2 u(x), $$ but it does not work with $H^1_0$, where $$ \|u(x)\|^2_{H^1_0} = \int\limits_0^l |u'(x)|^2 \operatorname{d\!}x $$