Consider the (fix) spatial domain $\Omega$ and the functional $F(x)$ as
$$F(x)=\int_{\Omega}f(x)d\Omega$$
where $x\in L^2(\Omega,\mathbb{R})$ and $f(x)$ is a smooth function. The first variational derivative with respect to $x$ is given by
$$\frac{\delta F(x)}{\delta x}=\frac{\partial f(x)}{\partial x}$$
Now consider that the domain is time varying according to a variable $\eta(t)$, i.e., $\Omega=\Omega(\eta(t))$ and
$$F(x,\eta)=\int_{\Omega(\eta)}f(x)d\Omega(\eta)$$
Then, how can we obtain the first variational derivative with respect to $\eta$?, i.e., $\frac{\delta F(x,\eta)}{\delta \eta}=?$
I try to solve it and obtain
$$\frac{\delta F(x,\eta)}{\delta\eta}=f(x)\frac{\partial \Omega(\eta)}{\partial \eta}$$
But I don't sure about it.