Suppose that $x\in \mathbb{H}$ is a local coordinate chart for $S^4$. Now we have the differential form $dx$, but know I want to compute what is $d\left(\frac{x}{|x|}\right)$ and I am not sure I am doing this the right way since non-commutativity , will it be $d\left(\frac{x}{|x|}\right)=\frac{dx\,|x|-x\,d|x|}{|x|^2}$? Also if anyone knows a reference where I can look this up I would appreciate it.
The result that I get is $\dfrac{2|x|^2x\,dx-dx \,\bar x-x\,d\bar x}{2|x|^3}$.
New edit : This is because I want to prove that $\omega_1=\frac{x\,d\bar x-dx\, \bar x}{2(1+|x|^2)},\omega_2=\frac{y\,d\bar y-dy \,\bar y}{2(1+|y|^2)}$ define local connection forms for a vector bundle over $S^4$ with transition function $g_{12}=\frac{x}{|x|}$. And then when trying to check that $\omega_2= g_{21}\omega_1 g_{12}+g_{21}d(g_{12})$ I am not getting the same thing so I am making some mistake.
$\omega_2=\frac{yd\bar y-dy\bar y}{2(1+|y|^2)}=\frac{x^{-1}-\bar x^{-1}d\bar x \bar x^{-1}+x^{-1}dxx^{-1}\bar x^{-1}}{2(1+|x^{-1}|^2)}$ $=\frac{ -d\bar x \bar x^{-1}+x^{-1}dx}{2(1+|x|^2)}$