Let $E,F$ be Banach spaces, $X$ open in $E^n$ and $f\colon X\to F$ be differentiable in $x_0\in X$. We know that for $v\in E^n\setminus\{0\}$ the directional derivative $$ \partial_v f(x_0) = \lim_{t\to 0}\frac{f(x_0+tv)-f(x_0)}{t}$$ exists and that $\partial_vf(x_0) =\partial f(x_0)v$. Does this propagate to higher order derivatives?
Let $f$ be twice differentiable in $x_0$. I am interested in if it is true that $$\partial^2f(x_0)(h,k) = \partial_k\partial_hf(x_0) = \partial(\partial f(\cdot)h)(x)k,\quad h,k\in E^n$$ (I think it is) and how I would prove it.