Consider the Lagrangian $L:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}:x,y,z\rightarrow L(x,y,z)$ and cost functional defined by the integral of a lagrangian $$J(y) := \int_{a}^bL(x,y(x),y'(x))dx$$ I would like to prove that the first variation obtained by the Gateaux derivative $$\delta J|_y(\eta)=\int_a^bL_y(x,y(x),y'(x))\eta(x) + L_z(x,y(x),y'(x))\eta'(x)dx$$ is also the first variation for the Frechet derivative, which is defined as $$J(y+\eta):=J(y) + \delta J|_y(\eta) + o(||\eta||)$$ for the 1-norm on the space of continuously differentiable functions $$||\eta||:= \max_{a\leq x\leq b}|\eta(x)| + \max_{a\leq x\leq b}|\eta'(x)|$$.
Basically, I'd like some help with proving $$\lim_{\eta\rightarrow 0}\frac{||\int_{a}^bL(x,y+\eta,y'+\eta') - L(x,y,y')-L_y(x,y,y')\eta + L_z(x,y,y')\eta'dx||}{||\eta||} = 0$$ possibly using only the definitions above. I'm new to calculus of variations and I'm not sure on how to expand the part coming from $J(y+\eta)$. Thank you in advance.