It's the chain rule. We have r_{ij} = \sqrt{w} where I've defined w as all those terms underneath the square root. So \frac{d r_{ij}}{dt} =\frac{d r_{ij}}{dw} \frac{dw}{dt} = \frac{1}{2 \sqrt{w}} \frac{dw}{dt} = \frac{1}{2 r_{ij}} \frac{dw}{dt} \,, ...

One simplification is to set \begin{align} a &= \text{xPos}_2 - \text{xPos}_1, \\ b &= \text{xVel}_2 - \text{xVel}_1, \\ c &= \text{yPos}_2 - \text{yPos}_1, \\ d &= \text{yVel}_2 - \text{yVel}_1. ...

Define u_r+\sqrt{d}v_r:=\left(u+\sqrt{d}v\right)^r, where u_r,v_r\in\mathbb{Z} and r\in\mathbb{N}_0. Then, you can see that v_{r+2}+av_{r+1}+bv_r=0 for some a,b\in\mathbb{Z} and for all r\in\mathbb{N}_0 ...

x\underbrace{\sqrt{x^2+y^2}}_{=|z|}-2x+y\underbrace{\sqrt{x^2+y^2}}_{=|z|}i-2yi+i=|z|(x+iy)-2(x+y)+i=z|z|-2z+i