Just to put my oar in, you know, I hope, that the dervative of sin(x) is cos(x) and the derivative of cos(x) is -sin(x).
From that, and , the derivative of tan(x) is .
Now, if y= arctan(x) then x= tan(y) so that . But so
That is, so .
That tells us that
and the general formula can be derived by substitution as others have shown.