The derivative of the Heaviside step function in $\mathbb{R}^n$

Question

Here is a problem I have:

Consider the Heaviside function $$H(x)=H(x_1)H(x_2)\cdots H(x_n), ~~~x\in\mathbb{R}^n$$ and prove that $$\dfrac{\partial^n H(x)}{\partial x_1\partial x_2\cdots\partial x_n}=\delta(x)$$ where $\delta(x)$ is the Dirac's delta function.

I know $H'(x)=\delta(x)$ in $\mathbb{R}$, but here $x\in\mathbb{R}^n$ how can I proceed for the $nth$ partial derivative of $H(x)$? Any help?

Answer 1

How about: $$\frac{\partial^n H(\mathbf x)}{\partial x_1\cdots \partial x_n} = \frac{\operatorname d H(x_1)}{\operatorname d x_1}\cdots \frac{\operatorname d H(x_n)}{\operatorname d x_n} = \delta(x_1)\cdots \delta(x_n),$$ which is the product of one-dimensional Delta distributions.