Heat equation and its gradient flow

Question

Let $u$ be the solution of the heat equation in $\Omega\subset \mathbb R$. So $u(x, t)$ is the solution of $$\frac{\partial}{\partial t}u(x,t)=\frac{1}{2}\frac{\partial^2}{\partial x^2}u(x, t), \qquad x\in \Omega, t\geq 0,\qquad (1)$$ with the boundary condition $\frac{\partial}{\partial x}u(x, t)=0, x\in\partial \Omega$. The associated energy is given by $$E(u)=\frac{1}{2}\int_{\Omega}\Big(\frac{\partial}{\partial x}u(x,t)\Big)^2dx$$ in the sense that (1) can be rewritten as $$\frac{\partial}{\partial t}u(x,t)=-\frac{d}{du}E(u)$$. My question is how can I obtain that $$\frac{d}{du}E(u)=-\frac{1}{2}\frac{\partial^2}{\partial x^2}u(x, t)?$$