Given y = (x*logx)^log(logx)

taking log on both sides, we get

      logy = log[(x*logx)^log(logx) ]

=> logy = log(logx)*log(x*logx)

Now, differentiate w.r.t. x, we get

      (1/y)*(dy/dx) = log(x*logx)*{(1/logx) * (1/x)} + log(logx)*[(1/x*logx)*{logx + x/x}]

=> (1/y)*(dy/dx) = log(x*logx)*(1/x*logx) + log(logx)*[(1/x*logx)*{logx + 1}]

=> (1/y)*(dy/dx) = (log(x*logx)/(x*logx) + {log(logx)*(logx + 1)/(x*logx)

=> (1/y)*(dy/dx) = (log(x*logx)/(x*logx) + {log(logx){1/x + 1/(x*logx)}

=> dy/dx = y[(log(x*logx)/(x*logx) + {log(logx){1/x + 1/(x*logx)}]

=> dy/dx = y{(log(x*logx)/(x*logx) + log(logx)/x + log(logx)/(x*logx)}

=> dy/dx = {(x*logx)^log(logx) }*{(log(x*logx)/(x*logx) + log(logx)/x + log(logx)/(x*logx)}