First, use the product rule:
#f'(x)=(1+e^(x^2))^(1/5)color(blue)(d/dx[x])+xcolor(green)(d/dx[(1+e^(x^2))^(1/5)]#
Find each derivative separately.
#color(blue)(d/dx[x]=1#
The following requires heavy use of the chain rule:
#color(green)(d/dx[(1+e^(x^2))^(1/5)])=1/5(1+e^(x^2))^(-4/5)color(red)(d/dx[1+e^(x^2)]#
#color(red)(d/dx[1+e^(x^2)])=e^(x^2)d/dx[x^2]=color(red)(2xe^(x^2)#
Plug this back in.
#color(green)(d/dx[(1+e^(x^2))^(1/5)])=(2xe^(x^2)(1+e^(x^2))^(-4/5))/5=color(green)((2xe^(x^2))/(5(1+e^(x^2))^(4/5))#
Finally, plug these in to find #f'(x)#.
#f'(x)=(1+e^(x^2))^(1/5)+(2x^2e^(x^2))/(5(1+e^(x^2))^(4/5))#
Multiply the first term for a common denominator.
#f'(x)=(5+5e^(x^2)+2x^2e^(x^2))/(5(1+e^(x^2))^(4/5))#
