I tried to understand the reasoning given in it but I couldn't understand it. It says that "as the gradient operation involves x and not the integration variable x', it can be taken outside the integral sign".
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You are essentially using Leibinz integral rule
$$ {\displaystyle {\frac {{\rm d}}{{\rm d}x}}\left(\int _{a}^{b}f(x,x')\,{\rm d}x'\right)=\int _{a}^{b}{\frac {\partial }{\partial x}}f(x,x')\,{\rm d}x'.} $$
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$\begingroup$ I know this but what does the statement mean that it does not involve integration variable x'? $\endgroup$ Commented Dec 27, 2018 at 12:31
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$\begingroup$ @HaaranAjgaonkar It means that you are integrating w.r.t $x'$, but taking the derivative w.r.t $x$, so the rule above applies $\endgroup$– caveracCommented Dec 27, 2018 at 12:40
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$\begingroup$ Thank you very much, I was a little bit confused by the language of the context. $\endgroup$ Commented Dec 27, 2018 at 12:45