We will differentiate #F# and wade our way slowly and methodically through finding the derivative of the right. First with the outermost function and chain rule:
#F(x)=f(color(blue)(xf(xf(x))))#
#F'(x)=f'(color(blue)(xf(xf(x))))(d/dxcolor(blue)(xf(xf(x))))#
Now applying the product rule:
#F'(x)=f'(xf(xf(x)))[f(xf(x))+x(d/dxf(xf(x)))]#
Reapplying the chain rule:
#F'(x)=f'(xf(xf(x)))[f(xf(x))+x(f'(xf(x))+(d/dx xf(x)))]#
#F'(x)=f'(xf(xf(x)))[f(xf(x))+xf'(xf(x))+x(d/dx xf(x))]#
Product rule once more:
#F'(x)=f'(xf(xf(x)))[f(xf(x))+xf'(xf(x))+x(f(x)+xf'(x))]#
We could simplify this a little more, but I'm dubious it would help. Evaluating at #x=1#:
#F'(1)=f'(f(f(1)))[f(f(1))+f'(f(1))+1(f(1)+f'(1))]#
#F'(1)=f'(f(2))[f(2)+f'(2)+2+4]#
#F'(1)=f'(3)(3+5+2+4)#
#F'(1)=6(14)#
#F'(1)=84#
