The Differential answer,i calculated, is not the same as the solution

Question

The question is on first picuture,and the part of solution is on the second picture.I am not quite understand the second picture meaning

(1)

(2)

The differential of $g(z)$ ,$g'(z)$, should be $\frac{-e^{-z}}{(1+e^{-z})^2}$,why is the $g'(z)$ equal to $a_i^{(l)}(1-a_i^{(l)})$?

Answer 1

$g(z)=\frac{1}{1+e^{-z}}$,then $$g'(z)=-(1+e^{-z})^{-2}(-e^{-z})=\frac{e^{-z}}{(1+e^{-z})^2}$$ Notice that there is no negative sign.

\begin{align} g'(z) &= \frac{e^{-z}}{(1+e^{-z})^2} \\ &=\frac{1}{1+e^{-z}}\cdot \frac{e^{-z}}{1+e^{-z}} \\ &= a\cdot \left(1 - \frac{1}{1+e^{-z}}\right)\\ &= a(1-a) \end{align}

where I let $a=g(z)$.