Vtyjkyuk

I have two 2-variable functions
$u=u(x,t), g=g(x,t)$

I want to simplify the expression:

$ fracpartialpartial t ( u*g)$

Where $*$ is the convolution of $u$ and $g$ with respect to $x$. My understanding of Wikipedia
https://en.wikipedia.org/wiki/Convolution#Properties
under the title Differentiation yields that

$ fracpartialpartial t ( u*g)= fracpartial upartial t*g $

But when I try to prove that I get:

$fracpartialpartial t ( u*g) =fracpartialpartial t int_-infty^infty u(x-alpha,t)g( alpha,t) dalpha =
int_-infty^infty fracpartialpartial t u(x-alpha,t)g(alpha,t) dalpha
=int_-infty^infty u'_t(x-alpha,t)g( alpha,t)+u(x-alpha,t)g'_t( alpha,t) dalpha =
u'_t*g+u*g'_t ne u'_t*g $

Where am I'm going wrong?

The formula you got from Wikipedia holds if are differentiating w.r.t. $x$, not w.r.t. $t$. For differentiation w.r.t $t$ what you have obtained is correct.

As the convolution is with respect to $x$ and the differentiation is with respect to $t$, you can't use the formulas from the Wikipedia article. Your derivation looks correct to me.