Given,  y = cos^-1 {(1 - x² )/(1 + x² )}

=> y = 2tan^-1 (x)                               {since 2tan^-1 (x) = cos^-1 {(1 - x² )/(1 + x² )}}

Now, differentiate w.r.t. x, we get

      dy/dx = d{2tan^-1 (x)}/dx

=> dy/dx = 2 * d{tan^-1 (x)}/dx

=> dy/dx= 2 * 1/(1 + x² )

=> dy/dx = 2/(1 + x² )