The problem statment. Let $f\in C^m(E)$, where $E$ is an open subset of $\Bbb{R^n}$. Fix $a\in E$, and suppose $x\in \Bbb{R^n}$ is so close to $0$ that the points $p(t) a+ tx$ lie in $E$ whenever $1\geq t\geq0$. Define $h(t)=f(p(t))$ for all $t\in \Bbb{R}$ for which $p(t)\in E$. Problem : For $m\geq k\geq1$, show that $h^{(k)}(t)=\sum(D_{i_1 ...i_k}f)(p(t))x_{i_1}...x_{i_k}$. The sum extends over all ordered k-tuples $(i_1...i_k)$ in which each $i_j$ is one of the integers $1,...,n$.
Using the chain rule repeatedly is recommended, so I computed $$h^{(1)}(t)=h'(t)=f'(p(t))\cdotp p'(t)=\sum_i^n(D_if)(p(t))\cdot x_i$$
But then $h^{(2)}=f''(p(t))\cdot p''(t)=0.$ It doesn't seem like this kind of process will lead towards the answer. Suggestions?