\mu)$ for the derivative w.r.t. the first argument, not $\partial_{u_\mu} {\mathcal{J}}(u_\mu, \mu)$ or $\partial_v a_\mu(u, p)$ for the derivative w.r.t. the second argument, not $\partial_p a_\mu(u, p)$. Note also that, due to Assumption \[asmpt:parameter\_separable\], we can exchange the order of differentiation w.r.t. $V$ and $\mathbb{R}^P$, i.e. $\partial_u\big(\partial_\mu {\mathcal{J}}(u, \mu)\cdot\nu\big)[w] = \partial_\mu\big(\partial_u {\mathcal{J}}(u, \mu)[w]\big)\cdot\nu$.

\[asmpt:differentiability\] We assume $a_{\mu}$, $l_{\mu}$ and ${\mathcal{J}}$ to be twice Fréchet differentiable w.r.t. $\mu$. This obviously requires that all parameter-dependent coefficient functions in Assumption [[\[asmpt:parameter\_separable\]]{}]{} are twice differentiable as well.

\[prop:gateaux\_wrt\_V\] Due to the (bi-)linearity of $a$, $l$, $j$ and $k$, we can immediately compute their derivatives w.r.t. arguments in $V$. For $u, v \in V$, $\mu \in {\mathcal{P}}$, the derivatives of $a$, $l$ and ${\mathcal{J}}$ w.r.t. arguments in $V$ in the direction of $w \in V$ are given, respectively, by $$\partial_u a_\mu(u, v)[w] = a_\mu(w, v),\hspace{1em}\partial_v a_\mu(u, v)[w] = a_\mu(u, w), \hspace{1em} \partial_v l_\mu(v)[w] = l_\mu(w), \hspace{1em}\partial_u {\mathcal{J}}(u, \mu)[w] = j_\mu(w) + 2 k_\mu(w, u).$$

We compute the partial derivatives of $a$ and $l$ w.r.t. the parameter by means of their separable decomposition.

\[rem:gateaux\_wrt\_P\] For $\mu \in {\mathcal{P}}$, $u, v, \in V$ the derivatives of $a$ and $l$ w.r.t. $\mu$ in the direction of $\nu \in 