This problem is in Chap.3 of Introduction to Matrix Analysis and Applications of Petz.
If $f (x) = x^k$ with $k \in \mathbb{N}$, verify that $$ f^{[n]}\left[x_{1}, x_{2}, \ldots, x_{n+1}\right] = \sum_{u_{1}, u_{2}, \ldots, u_{n+1} \geq 0 \atop u_{1}+u_{2}+\ldots+u_{n+1} = k-n} x_{1}^{u_{1}} x_{2}^{u_{2}} \cdots x_{n}^{u_{n}} x_{n+1}^{u_{n+1}} $$
I want to use inductive reasoning to proof this problem.
According the divided differences. $$f^{[0]}\left[x_{1}\right]:=f\left(x_{1}\right), \quad f^{[1]}\left[x_{1}, x_{2}\right]:=\frac{f\left(x_{1}\right)-f\left(x_{2}\right)}{x_{1}-x_{2}}$$ and recursively for $n = 2, 3,...$, $$f^{[n]}\left[x_{1}, x_{2}, \ldots, x_{n+1}\right]:=\frac{f^{[n-1]}\left[x_{1}, x_{2}, \ldots, x_{n}\right]-f^{[n-1]}\left[x_{2}, x_{3}, \ldots, x_{n+1}\right]}{x_{1}-x_{n+1}}$$ It can obtain following equations.
$f(x_1)=x_1^k$ fits the equation of the question.
However, $f'(x_1,x_2)=\frac{x_1^k-x_2^k}{x_1-x_2}$ is seems not follow the equation.
I'm currently stuck here.
Hint: Compute $(x-y)\sum_{i=0}^k x^iy^{k-i}$