Taylor expansion of retarded vector potential

Question

I am currently trying to understand the Taylor expansion in radiation zone of the vector potential for a general time dependence. I know that the retarded potential is given by:
$$textbf{A}(textbf{r},t) = frac{mu_0}{4pi}intfrac{textbf{J}({textbf{r}'}, t_r)}{R},mathrm dV',$$
where $t_r = t - R/c$ and $R = |textbf{r}-textbf{r}'|$. Since we are in radiation zone we can approximate $R approx r$ and then Taylor expand the current density around $t' = t - r/c$. By just including the first two terms of the current density we get:
$$textbf{A}(textbf{r},t) = frac{mu_0}{4pi r}inttextbf{J}({textbf{r}'}, t') + dot{textbf{J}}({textbf{r}'}, t')frac{textbf{r}'cdot hat{textbf{r}}}{c},mathrm dV',$$
where the dot over $textbf{J}$ represents the derivative with respect to $t'$. I have seen multiple documents where after this step, they write the integral as following:
$$textbf{A}(textbf{r},t) = frac{mu_0}{4pi r}inttextbf{J}({textbf{r}'}, t'),mathrm dV' + frac{mu_0}{4pi r} frac{partial}{partial t'}inttextbf{J}({textbf{r}'}, t')frac{textbf{r}'cdot hat{textbf{r}}}{c},mathrm dV'.$$
My question is: How one can put the partial derivative outside the integral? In further calculations, one uses that $textbf{r}'$ is dependent on $t'$ so should it not be any kind of product rule when one is putting the derivative outside the integral?

tmsn · Answer

Note that $t'$ is not the retarded time corresponding to the position $mathbf r'$.  Instead, $t'$ is the retarded time corresponding to the position $mathbf r$: $ t' = t - r/c  $. So, $t'$ as defined here is independent of $mathbf {r}'$.

$large int mathbf {dot J}(mathbf r', t') frac{mathbf r' cdot hat {mathbf {r}} }{c}dV' = intleft[lim limits_{hto 0} frac{mathbf J(mathbf r', t'+h)-mathbf J(mathbf r', t')}{h} right] frac{mathbf r' cdot hat {mathbf {r}} }{c} dV' = $

$large lim limits_{hto 0} frac{ int mathbf J(mathbf r', t'+h)frac{mathbf r' cdot hat {mathbf {r}} }{c}dV' -int mathbf J(mathbf r', t')frac{mathbf r' cdot hat {mathbf {r}} }{c}dV'   }{h}=  $

$large frac{partial}{partial t} left[int mathbf J(mathbf r', t)frac{mathbf r' cdot hat {mathbf {r}} }{c}dV'  right]_{t=t'} equiv frac{partial}{partial t} int mathbf J(mathbf r', t')frac{mathbf r' cdot hat {mathbf {r}} }{c}dV'$

Taylor expansion of retarded vector potential

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