Define the Hermite polynomial $H_n$ by $$H_n=(-1)^n e^{x^2}\frac{d^2 e^{-x^2}}{dx^n}.$$
Now let $x\in\mathbb{R}^d$, $\alpha\in\mathbb{N}_0^d$ be a multi-index and $\partial^\alpha=\frac{\partial^{\alpha_1}}{\partial x^{\alpha_1}}...\frac{\partial^{\alpha_d}}{\partial x^{\alpha_d}}$.
Now I have to show that $$\partial^\alpha e^{-x\cdot x}=(-1)^{|\alpha|}\prod_{j=1}^d H_{\alpha_j}(x_j)e^{-x\cdot x}$$
Honestly I dont really know how to start, so help is very much appreciated!
Thank you.
It follows from the simple factorization property $$ \frac{\partial^2}{\partial x\partial y} f(x)g(y)=f'(x)g'(y)\ . $$