Find the derivative

take logarithm to both side
log h(x) = xlog (log 2x)
differentiate both w.r.t x
1/h(x) * h'(x) = log(log2x)+x*1/(log2x)*1/(2x)*2
1/h(x) * h'(x) =log(log2x)+1/(log2x)
h'(x)= {log(log2x)+1/(log2x)}*{(log2 x)^x}