FEniCS’s UFL includes the derivative(E,eta,v) function, which implements Gateaux differentiation of a functional E with respect to a Function, eta, in the direction v. In mathematical notation, that is
D_vE[\eta] = \left.\frac{d}{d\epsilon}\left(E[\eta + \epsilon v]\right)\right\vert_{\epsilon=0}\text{ ,}
where \epsilon is a scalar. If the direction v is left out (i.e., derivative(E,eta)), then it is taken to be an arbitrary TestFunction in the same FunctionSpace as eta. Based on the Wikipedia article on “functional derivatives”, it seems that the notation from your post is a common alternative way of writing the Gateaux derivative, i.e.,
D_vE = \int_V\frac{\delta E}{\delta \eta} v\,dV\text{ .}