Theorem If $i,k$ are multi-indices in $\mathbb{N}^{n}$, and $x=(x_{1},\ldots,x_{n})$, then

Proof. The proof follows from the corresponding rule for the ordinary derivative; if $i,k$ are in $0,1,2,\ldots$, then

Suppose $i=(i_{1},\ldots,i_{n})$, $k=(k_{1},\ldots,k_{n})$, and $x=(x_{1},\ldots,x_{n})$. Then we have that

For each $r=1,\ldots,n$, the function $x_{r}^{{k_{r}}}$ only depends on $x_{r}$. In the above, each partial differentiation $\partial/\partial x_{r}$ therefore reduces to the corresponding ordinary differentiation $d/dx_{r}$. Hence, from equation 1, it follows that $\partial^{i}x^{k}$ vanishes if $i_{r}>k_{r}$ for any $r=1,\ldots,n$. If this is not the case, i.e., if $i\leq k$ as multi-indices, then for each $r$,

and the theorem follows. $\Box$