Problem

f(x)= (((cos(x)/tan(x))^1/2)^1/2)^1/2

   =(cos(x)/tan(x))^1/8

Solution

f´(x)= 1/8 * (cos(x)/tan(x))^{1/8 - 1} * ((-sin(x) * tan(x)+ cos(x)* 1/(cos(x))²

    = 1/8 * (cos(x)/tan(x))^{1/8 - 1} * ((-sin(x) * tan(x)+ 1/cos(x))