I'm taking a course on Neural Networks.
one of the questions on our exam will (likely) be:
Show by calculation that the derivative of the Fermi function (logistic function) can be expressed by the function itself.
Accordingly I've consulted with my friend, he proposed me this as a solution:
Step 1
$ f(x) = 1/(1+e^{-x}) $
→
$ f'(x) = \frac{d}{dx} (1 + e^{-x})^{-1}$
To interject here, the $\frac{d}{dx}$ term is basically meaningless- it just indicates that we're taking a derivative- isn't it?
Step 2
$ f'(x) = \frac{d}{dx} (1 + e^{-x})^{-1}=$
$ = -1 ( (1 + e^{-x})^{-2} ) ( (0 + e^{-x}) (-1) )=$
Step 3
$= -1 ( (1 + e^{-x})^{-2} ) ( (0 + e^{-x}) (-1) )=$
$ = e^{-x} / (1 + e^{-x})^2 =$
Step 4
$ = e^{-x} / (1 + e^{-x})^2 =$
$ = (1+ e^{-x} -1) / (1 + e^{-x})^2 =$
Step 5
$ = (1+ e^{-x} -1) / (1 + e^{-x})^2 =$
$ = ( 1 / ( 1 + e^{-x} ) ) - ( 1 / (1 + e^{-x})^2 )= $
Step 6
$ = ( 1 / ( 1 + e^{-x} ) ) - ( 1 / (1 + e^{-x})^2 ) =$
$ = f(x) - ( f(x) )^2 =$
Step 7
$ = f(x) - ( f(x) )^2 =$
$= f(x) ( 1 - f(x) )$
So essentially what I would like to know is: is that correct?
But also- I would like to know how we arrived to that solution, is it possible that someone could explain this to me in English as if I were a 5 years old child?