#y = x^(e^(x^2))#
#ln y = e^(x^2)lnx#
Apply implicit diferentiation
#1/ydy/dx = d/dx(e^(x^2)lnx)#
#= e^(x^2).d/dx lnx + d/dx e^(x^2) lnx# [Product rule]
#= e^(x^2)1/x + e^(x^2) . 2x.lnx# [Standard differential and Chain rule]
#= e^(x^2)/x (2x^2lnx +1)#
#:. dy/dx = x^(e^(x^2)) xx e^(x^2)/x (2x^2lnx +1)# [Replace #y#]
#= x^(e^(x^2))/x e^(x^2) (2x^2lnx +1)#
#=x^(e^(x^2)-1).e^(x^2)(2x^2lnx+1)#