I need help understanding thsi calc 2 problem?

We are doing integration by parts, and I was going back over and practicing my lecture notes when I came across the problem “find the integral using IBP for arctan(x)dx... I know we start by IBP substitution where u=arctanx, dv=1 and v=x...I don’t understand why we use du/dx=1dx/1+x^2 for du? Further on I use this and I start to have to derive x/1+x^2.. why does this (after u sub, u=1+x^2)  simplify to 1/u and du/2 when we are supposed to be deriving? If someone could work this out for me that would be amazing! Thank you!