We know that the Fréchet differential $DF(u,\delta)$ of a functional $F:V\to V$ is satisfied (cf. Wiki) $$ \lim_{\delta\to 0}{\dfrac{\left\|F(u+\delta)-F(u)-DF(u,\delta)\right\|_V}{\left\|\delta\right\|}}=0 \qquad (1) $$ where $V$ is a functional space, for example, $V=H_0^1(\Omega)$ with $\Omega = \{x=(x_1,x_2):x_i\in \mathbb{R}\}$.
Or we have the alternative definition (cf. EncyclopediaOfMath): $$F(u+\delta)=F(u)+DF(u,\delta)+o(\delta)$$ where $\lim_{\left\|\delta\right\|\to 0}{\dfrac{\left\|o(\delta)\right\|}{\left\|\delta\right\|}}=0$.
The question is how to calculate the Fréchet differential of a given function/functional? If we only need to check whether some $DF(u,\delta)$ is Frechet differential of $F$ or not, that is easier because we only need to check (1) but in this case, I need to find the differential.
For example, find the Fréchet differential of $F$ given by $$F(u)=\int_{\Omega}{K(u)\nabla u\cdot\nabla v}, \qquad \forall v\in V$$ where $K(u)$ which is a nonlinear functional w.r.t $u$ has some neccessary conditions. If we use Gâteaux differential (cf. Wiki), we will get the result: $$DF(u,\delta)=\int_{\Omega}{K'(u)\delta\nabla u\cdot\nabla v}+\int_{\Omega}{K(u)\nabla\delta\cdot\nabla v}$$
You just write down $F(u+\delta)$ and subtract $F(u)$. Then you cross your fingers and hope to isolate a term that is linear in $\delta$. Finally, you try to prove that this linear term is the differential of $F$. And do not forget that there rules of differentiation, similar to those you learn in a calculus course.
Of course, you may also try to compute the Gateaux derivative, and then prove its continuity.