Sorry, but that is wrong. If we have:
A = sigmoid(Z)
Then
g'(Z) = A * (1 - A)
But you don’t really need that in order to compute dZ^{[L]}. They have you the shortcut that comes from that together with the formula for dA^{[L]}.
dZ^{[L]} = A^{[L]} - Y
Here’s another recent thread that may be worth a look.