Let y=(tan(2x))^x, then ln(y)=xln(tan(2x)).
Differentiating we get: (1/y)dy/dx=2x(sec(2x))^2/tan(2x)+ln(tan(2x)),
so dy/dx=2xy(sec(2x))^2/tan(2x)+yln(tan(2x))
y/tan(2x)=(tan(2x))^(x-1) from the original equation.
dy/dx=2x(1+(tan(2x))^2)(tan(2x))^(x-1)+((tan(2x))^x)(ln(tan(2x))).
tan(2x)=2x when x is small, so dy/dx=2x*(2x)^-1+0*(-X) where X is very large; therefore dy/dx=2x/2x=1, and the limit as x approaches 0 is 1.