I'm not going to include a lot of words in this explanation, since it's mostly just math. There's a lot of special rule usage, but here we go:
#color(white)=d/dx[arctan(x/(1+sqrt(1-x^2)))]#
Chain rule (derivative of #arctan# is #1/(1+x^2)#):
#=1/(1+(x/(1+sqrt(1-x^2)))^2)*d/dx[x/(1+sqrt(1-x^2))]#
#=1/(1+x^2/(1+sqrt(1-x^2))^2)*d/dx[x/(1+sqrt(1-x^2))]#
#=1/(1+x^2/(1+2sqrt(1-x^2)+1-x^2))*d/dx[x/(1+sqrt(1-x^2))]#
#=1/(1+x^2/(2+2sqrt(1-x^2)-x^2))*d/dx[x/(1+sqrt(1-x^2))]#
#=1/((2+2sqrt(1-x^2)-x^2)/(2+2sqrt(1-x^2)-x^2)+x^2/(2+2sqrt(1-x^2)-x^2))*d/dx[x/(1+sqrt(1-x^2))]#
#=1/((2+2sqrt(1-x^2)-x^2+x^2)/(2+2sqrt(1-x^2)-x^2))*d/dx[x/(1+sqrt(1-x^2))]#
#=1/((2+2sqrt(1-x^2))/(2+2sqrt(1-x^2)-x^2))*d/dx[x/(1+sqrt(1-x^2))]#
#=(2+2sqrt(1-x^2)-x^2)/(2+2sqrt(1-x^2))*d/dx[x/(1+sqrt(1-x^2))]#
Quotient rule:
#=((2+2sqrt(1-x^2)-x^2)/(2+2sqrt(1-x^2)))*(d/dx[x](1+sqrt(1-x^2))-x*d/dx[1+sqrt(1-x^2)])/(1+sqrt(1-x^2))^2#
#=((2+2sqrt(1-x^2)-x^2)/(2+2sqrt(1-x^2)))*(1+sqrt(1-x^2)-x*d/dx[sqrt(1-x^2)])/(1+sqrt(1-x^2))^2#
#=((2+2sqrt(1-x^2)-x^2)/(2+2sqrt(1-x^2)))*(1+sqrt(1-x^2)-x*d/dx[sqrt(1-x^2)])/(1+2sqrt(1-x^2)+1-x^2)#
#=((2+2sqrt(1-x^2)-x^2)/(2+2sqrt(1-x^2)))*(1+sqrt(1-x^2)-x*d/dx[sqrt(1-x^2)])/(2+2sqrt(1-x^2)-x^2)#
#=(color(red)cancelcolor(black)(2+2sqrt(1-x^2)-x^2)/(2+2sqrt(1-x^2)))*(1+sqrt(1-x^2)-x*d/dx[sqrt(1-x^2)])/color(red)cancelcolor(black)(2+2sqrt(1-x^2)-x^2)#
#=(1+sqrt(1-x^2)-x*d/dx[sqrt(1-x^2)])/(2+2sqrt(1-x^2))#
Chain rule (derivative of #sqrtx# is #1/(2sqrtx)#):
#=(1+sqrt(1-x^2)-x*(1/(2sqrt(1-x^2))*d/dx[1-x^2]))/(2+2sqrt(1-x^2))#
#=(1+sqrt(1-x^2)-x*(1/(2sqrt(1-x^2))*-2x))/(2+2sqrt(1-x^2))#
#=(1+sqrt(1-x^2)-x*(-2x)/(2sqrt(1-x^2)))/(2+2sqrt(1-x^2))#
#=(1+sqrt(1-x^2)+x^2/sqrt(1-x^2))/(2+2sqrt(1-x^2))#
#=(1+sqrt(1-x^2)+x^2/sqrt(1-x^2))/(2+2sqrt(1-x^2))color(red)(*sqrt(1-x^2)/sqrt(1-x^2))#
#=(sqrt(1-x^2)+1-x^2+x^2)/(2sqrt(1-x^2)+2(sqrt(1-x^2))^2)#
#=(sqrt(1-x^2)+1)/(2sqrt(1-x^2)+2(sqrt(1-x^2))^2)#
#=(sqrt(1-x^2)+1)/((2sqrt(1-x^2))(1+sqrt(1-x^2))#
#=color(red)cancelcolor(black)(sqrt(1-x^2)+1)/((2sqrt(1-x^2))color(red)cancelcolor(black)((1+sqrt(1-x^2)))#
#=1/(2sqrt(1-x^2))#
Finally, that's it. Hope this helps!
