Please differentiate the following:

Let 2^x = tanA

So , the equation becomes

cos^-1( 2.tanA / (1+tan²A) ) = cos^-1(sin2A) = pie/2 - sin^-1(sin2A) = pie/2 - 2A = pie - 2tan^-12^x

the derivative of the above function is

{-2/(1+4^x)} .2^x log2