Calc of variation - the variation of a derivative?

[pellman]

If we take the variation of a functional of some function \phi(x_1,...,x_n) with \partial_{j}\phi being the partial deriviative of phi with respect x_j, when is it ok to set \delta \partial_j \phi equal to \partial_j (\delta\phi)?

 

[pellman]

Maybe this equality holds whenever we are not varying the surface of the region of integration? Can someone confirm?