Is the following differentiating under the integral sign correct?

Question

Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial u}-\frac{\partial }{\partial x}\frac{\partial f}{\partial u_x}+\left(\frac{\partial }{\partial x}\right)^2\frac{\partial f}{\partial u_{xx}}+...$$ and $$f[u]=\int_\mathbf{R} \phi(u,u_x)dx$$ What then is $$\frac{\partial }{\partial x}\frac{\delta f[u]}{\delta u(x)}$$?

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My guess would be

$$\frac{\partial }{\partial x}\int_\mathbf{R} \frac{\partial \phi}{\partial u}-\frac{\partial }{\partial x}\frac{\partial \phi}{\partial u_x}dx$$

Then $$\left[\frac{\partial \phi}{\partial u}-\frac{\partial }{\partial x}\frac{\partial \phi}{\partial u_x}\right]_{-\infty}^{\infty}$$ Would that be right?

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Anybody?

Answer 1

Have been informed that I misunderstood the notation in my notes. The problem is resolved now! Sorry about that!