How exactly is he using the mean value theorem to get, for example, $u(x+s,y+t) - u(x,y+t) = u_x(x+s_1,y+t)s$? It appears that he is treating $s,y,$ and $y$ as constants; but I can't figure out how he is applying it.
@user193319 If we simply fix $y+t$ and view this as a function of the first variable, we get the expression you posted.
I mean, consider the function $f(x)=u(x,y+t)$ where $y,t\in\mathbb R$ are some fixed constant.
Then this expression simply says that $f(x+s)-f(x)=f'(x+s_1)s$.
Or, if you prefer this form, you can denote $x_1=x$ and $x_2=x+s$ and hit is just $f(x_2)-f(x_1)=f'(t)(x_2-x_1)$ for some $t\in(x_1,x_2)$.
