Higher derivatives of multilinear map

Question

Let, $E,F$ are two banach spaces and $V$ be an open subset of $E.$ $f_1,f_2:V\to F$ be twice continuously differentiable function.Let, $H$ be a multilinear map from $F\times F$ to a banach space $G$ and $\phi:U\to G$ defined as $\phi(x)= H(f_1(x),f_2(x)).$Now ,compute $D^2 \phi.$

I can compute $D^1(\phi)$ and that is, $D^1\phi(x)(z)= H(Df_1(x)(z),f_2(x))+H(f_1(x),Df_2(x)(z))\text{ where}~z\in E.$

Now, to compute the second derivative I am facing problem. Any help?

Answer 1

Why don't you continue applying the same rule to each term,

$D^1\phi(x)(z)= H(Df_1(x)(z),f_2(x))+H(f_1(x),Df_2(x)(z))$

$D^2\phi(x)(z,w)=\\= H(D^2f_1(x)(z,w),f_2(x)) + H(Df_1(x)(z),Df_2(x)(w))\\ + H(Df_1(x)(w),Df_2(x)(z)) + H(f_1(x),D^2f_2(x)(z,w))$