Banach space norm induced by inner product; differentiability

Question

Let $\left( X,\left\| \cdot \right\| \right)$ be a Banach space where the norm $\left\| \cdot \right\|$ is induced by an inner product $\langle \cdot ,\cdot \rangle$.
Let $f:X\times X\rightarrow X$, $f\left( x,y\right) =\langle y,x\rangle x$.

Show that f is differentiable in every point and calculate the total derivative.

I tried to show that all directional derivatives exist and are continuous but that didn‘t work. Is there a better way?

Answer 1

For Frechet differentiability, start by computing $$ f(x+h,y+k)-f(x,y)=\langle y+k,x+h\rangle(x+h)-\langle y,x\rangle x=\dots $$ and show it is "linear in $(h,k)$" + smaller.