Approximation of Sobolev function with smooth functions.

I have the following question regarding the following steps : 1) Why can i move the derivative $D^\alpha$ inside the integral sign.

2) I didn't understand why $D^\alpha$ changes to $D_x^\alpha$ and then to $D_y^\alpha$ ?

Thank you for ur explanation in advance.

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For part (2), the subscript indicates the point at which the derivative acts. So $$D^\alpha_x = \frac{\partial^{k_1}}{\partial x_1^{k_1}}\frac{\partial^{k_2}}{\partial x_2^{k_2}}\cdots\frac{\partial^{k_n}}{\partial x_n^{k_n}},$$ while $$D^\alpha_y = \frac{\partial^{k_1}}{\partial y_1^{k_1}}\frac{\partial^{k_2}}{\partial y_2^{k_2}}\cdots\frac{\partial^{k_n}}{\partial y_n^{k_n}},$$ with $x=(x_1,x_2,\dots,x_n)$, $y=(y_1,y_2,\dots,y_n)$ and $|\alpha|=k_1+k_2+\dots+k_n$. No subscript is used in the first line as the function depends on $x$ only. That said, there would be nothing incorrect in using $D^\alpha_x$ instead of $D^\alpha$ in the first line.