Let `y=cos^(-1)((x-x^(-1))/(x+x^(-1)))=cos^(-1){((x-(1)/(x)))/((x+(1)/(x)))}=cos^(-1)((x^(2)-1)/(x^(2)+1)).`
Putting `x=tan theta`, we get
`y=cos^(-1)((tan^(2)theta-1)/(tan^(2)theta+1))=cos^(-1)(-cos2 theta)`
`=cos^(-1){cos(pi-2 theta)}=pi-2 tan^(-1)x.`
`therefore(dy)/(dx)=(d)/(dx)(pi-2 tan^(-1)x)=(d)/(dx)(pi)-2.(d)/(dx)(tan^(-1)x)`
`=(0-(2)/(1+x^(2)))=(-2)/((1+x^(2))).`
