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?