Consider a function : $f(x^{1} \dots x^{n}) ($$x^{i}- i$-th coordinate $) \in C^{\infty}$ and $P$ such point : $\forall 0<l <k$ $\dfrac{\partial ^{l} f}{\partial x^{i_{1}} \dots \partial x^{i_{l}}}$ is zero at point $P$.
Now consider $A_{i_1 \dots i_k} =\dfrac{\partial ^{k} f}{\partial x^{i_{1}} \dots \partial x^{i_{k}}}$ we want to prove that it's a tensor. To prove this we need to check tensor-law: $A_{i'_1 \dots i'_k} =\dfrac{\partial x^{i_{1}}}{\partial x^{i'_{1}}} \dots \dfrac{\partial x^{i_{k}}}{\partial x^{i'_{k}}} \dfrac{\partial ^{k} f}{\partial x^{i_{1}} \dots \partial x^{i_{k}}}$.
I thought about considering $f = \sum_{n_1 \dots j_n} c(j_1 \dots j_n) x_{i_1}^{j_1} \dots x_{i_n}^{j_n}$ then condition about derivatives gives us some properties about coefficients. Maybe there is a better way to prove it?