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Let $X, X_1, X_2, Y $ be normed Vector spaces , $B : X_1 \times X_2 $ a bounded bilinear function and $f_i : X \longrightarrow X_i$ differentiable.

How do you show that the function $H:X \longrightarrow Y , H(x) := B(f_1(x),f_2(x)) $ is differentiable and how does the derivative look like?

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It's the composition of differentiable functions, which implies it is differentiable. It's derivative in direction $v$ is

$$DH (x) v= B(f_1(x),Df_2(x)v) + B(Df_1(x)v, f_2(x))$$

which follows from the fact that the derivative of the bilinear map $$ (v,w) \mapsto B(v,w) $$ is given by $$ DB(v,w) (e,f) = B(v,f)+ B(e,w)$$

This latter statement follows from the definition of the derivative from the following equality:

$$B(v+e, w+f) = B(v,w) + B(v,f) + B(e,w) + B(e,f) \\ = B(v,w) + DB(v,w)(e,f) + o(e,f)$$ (see, e.g., bilinear map and differentiability, if in doubt)

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