I would like to prove $
\nabla \varphi = lim _{\int dv \rightarrow 0} \frac{\int \varphi d \vec \delta }{\int dv}
$

with Gauss's theorem. According to Gauss's theorem, we know $
\int _{s} \varphi d \vec \delta = \int _{v} \nabla \varphi dv
$

Now if I want to prove it, I should write
$
\int _{v} \nabla \varphi dv = \nabla \varphi \int _{v} dv
$

Is this correct? How is that possible?