dy/dx=(-1/(1+4/x^2))(-2/x^2)+1/(2(1+x^2/4))=2/(x^2+4)+2/(x^2+4)=4/(x^2+4) (differentiation by parts)
The derivative of tan^-1(u)=1/(1+u^2); derivative of cot^-1(u)=-1/(1+u^2); derivative of 2/x=-2/x^2 and derivative of x/2=1/2. Differentiation by parts: dy/dx=dy/du * du/dx.