Suppose that $f:\mathbb{R}^n\to\mathbb{R}$ with $f(0)=0$ satisfying $$f(x)=o(\|x\|^m)\,,\,x\to0$$ If all the $m$-order partial derivatives exist, prove that the partial derivatives equal $0$ at the origin (or give a counterexample).
My attempt
Put $\phi(s)=f(sx_i)$. Thus $\phi$'s $m$-order derivative exists and $\phi(s)=o(s^m)$($s\to0$).
It implies that for $1\le k\le m$ we have $$\frac{\partial^kf}{\partial x_i^k}(0)=\phi^{k}(0)=0$$
But how to deal with something like $\frac{\partial^2f}{\partial x_1 \partial x_2}$? Any hints? Thanks in advance!