If y=exp(-exp(-x)), then what is the maximum value of dy/dx?

Question

Please answer for my question.

Answer 1

Let z=-e^-x, then dz/dx=e^-x.

y=e^z, dy/dx=e^zdz/dx=e^ze^-x=(e^-x)e^(-e^-x).

When dy/dx=0, x→∞ and e^-x→0 and e⁰=1, so dy/dx≈e^-x→0.

So there is no value of x which makes dy/dx=0.

When x=5, then z=-e^-5~-0.0067 and e^z=e^-0.0067=0.993~1. 

As x increases z→0, so y=e^z→1, an asymptote.

The maximum value of y=1 is never attained but is the limit as x→∞.

When x=0, z=-1 and y=e^-1=0.37 approx. when x<0, z<-1, y=e^z<0.36.

As x→-∞, z→-∞ and e^z→0. This shows that 0<y<1.