Question about DGLAP evolution equation

I am reading chapter 32.2 of Schwartz’s QFT book, where he defines the renormalized PDFs $f_i(x, mu)$. This leads to an equation (32.48), which relates PDFs at different scales $mu, mu_1$:
$f_i(x,mu_1) = f_i(x,mu) + frac{alpha_s}{2pi} int_x^1 frac{dxi}{xi} f_i(xi, mu_1) P_{qq}(frac{x}{xi}) ln(frac{mu_1^2}{mu^2})$
When I apply $mu frac{d}{dmu}$ to this equation I get:
$(1)~mu frac{d}{dmu}f_i(x,mu) = frac{alpha_s}{pi} int_x^1 frac{dxi}{xi} f_i(xi, mu_1) P_{qq}(frac{x}{xi})$.
However according to the book the correct equation is:
$(2)~mu frac{d}{dmu}f_i(x,mu) = frac{alpha_s}{pi} int_x^1 frac{dxi}{xi} f_i(xi, mu) P_{qq}(frac{x}{xi})$.
I am very confused about this. My questions is:

Why do we have $mu$ in the argument of $f_i$ in the RHS of equation $(2)$?

Doesn’t it follow that $frac{alpha_s}{pi} int_x^1 frac{dxi}{xi} f_i(xi, mu) P_{qq}(frac{x}{xi})$ is independent of $ mu$, since equation $(1)$ should be valid for any $mu_1$? This doesn’t make a lot of sense to me.
My other thought was that in order for the perturbation expansion to be good we have to choose $mu_1 sim mu$ so that the logarithm is small. So we can get from $(1)$ to $(2)$ by saying that they are approximately equal?