Let $f\in C^\infty(\mathbb R^{d_1}\times\mathbb R^{d_2})\,\cap\, W^{\infty,p}(\mathbb R^{d_1}\times\mathbb R^{d_2})$ for all $1<p<p_0$. Consider the functions $$ F(x_2) \,=\, \int_{\mathbb R^{d_1}} f(x_1,x_2) \,dx_1$$ $$ G(x_2) \,=\, \int_{\mathbb R^{d_1}} \frac{\partial f}{\partial x_2}(x_1,x_2) \,dx_1$$

Are extra conditions really needed in order to have $F(x_2),\,G(x_2)$ well defined and existence of $$\frac{\partial F}{\partial x_2}(x_2)=G(x_2)$$ for all $x_2\in\mathbb R^{d_2}$? Is it simpler if I require only absolute continuity of $F$ and the previous equality to hold only for almost every $x_2$?