In my AI textbook there is this paragraph, without any explanation.
The sigmoid function is defined as follows
$$\sigma (x) = \frac{1}{1+e^{-x}}.$$
This function is easy to differentiate because
$$\frac{d\sigma (x)}{d(x)} = \sigma (x)\cdot (1-\sigma(x)).$$
It has been a long time since I've taken differential equations, so could anyone tell me how they got from the first equation to the second?
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$\begingroup$Let's denote the sigmoid function as $\sigma(x) = \dfrac{1}{1 + e^{-x}}$.
The derivative of the sigmoid is $\dfrac{d}{dx}\sigma(x) = \sigma(x)(1 - \sigma(x))$.
Here's a detailed derivation:
$$ \begin{align} \dfrac{d}{dx} \sigma(x) &= \dfrac{d}{dx} \left[ \dfrac{1}{1 + e^{-x}} \right] \\ &= \dfrac{d}{dx} \left( 1 + \mathrm{e}^{-x} \right)^{-1} \\ &= -(1 + e^{-x})^{-2}(-e^{-x}) \\ &= \dfrac{e^{-x}}{\left(1 + e^{-x}\right)^2} \\ &= \dfrac{1}{1 + e^{-x}\ } \cdot \dfrac{e^{-x}}{1 + e^{-x}} \\ &= \dfrac{1}{1 + e^{-x}\ } \cdot \dfrac{(1 + e^{-x}) - 1}{1 + e^{-x}} \\ &= \dfrac{1}{1 + e^{-x}\ } \cdot \left( \dfrac{1 + e^{-x}}{1 + e^{-x}} - \dfrac{1}{1 + e^{-x}} \right) \\ &= \dfrac{1}{1 + e^{-x}\ } \cdot \left( 1 - \dfrac{1}{1 + e^{-x}} \right) \\ &= \sigma(x) \cdot (1 - \sigma(x)) \end{align} $$
$\endgroup$ 11 $\begingroup$Consider$$ f(x)=\dfrac{1}{\sigma(x)} = 1+e^{-x} . $$Then, on the one hand, the chain rule gives$$ f'(x) = \frac{d}{dx} \biggl( \frac{1}{\sigma(x)} \biggr) = -\frac{\sigma'(x)}{\sigma(x)^2} , $$and on the other hand,$$ f'(x) = \frac{d}{dx} \bigl( 1+e^{-x} \bigr) = -e^{-x} = 1-f(x) = 1 - \frac{1}{\sigma(x)} = \frac{\sigma(x)-1}{\sigma(x)} . $$Equate the two expressions, and voilà!
(Cf. also this answer.)
$\endgroup$ 3 $\begingroup$Note that from your given equation,
$(1+e^{-x})\sigma=1$
$\Rightarrow -e^{-x}\sigma+(1+e^{-x})\frac{d\sigma}{dx}=0$ (differentiating using product rule)
$\Rightarrow \frac{d\sigma}{dx}=\sigma.\frac{e^{-x}}{(1+e^{-x})}=\sigma.\frac{(1+e^{-x})-1}{(1+e^{-x})}=\sigma.\left[1-\frac{1}{(1+e^{-x})}\right]=\sigma.(1-\sigma)$
$\endgroup$ 0 $\begingroup$Since $\sigma(x)$ is a composite function, firstly we need to use chain rule to dig down to the x term, then we can factor back to the $\sigma(x)$ fuction:$$ \begin{align} \frac{d}{dx}\sigma(x) &= (\frac{1}{1+e^{-x}})' \\ &= -\frac{1}{(1+e^{-x})^{2}} \cdot (1) \cdot -e^{-x} \\ &= \frac{e^{-x}}{(1+e^{-x})^{2}}, \\ \because \sigma(x) &= \frac{1}{1+e^{-x}}, \\ e^{-x} &= \frac{1 - \sigma(x)}{\sigma(x)}, \\ 1+e^{-x} &= \frac{1}{\sigma(x)}; \\ \therefore \frac{d}{dx}\sigma(x) &= \frac{\frac{1 - \sigma(x)}{\sigma(x)}}{(\frac{1}{\sigma(x)})^{2}} \\ &= (1 - \sigma(x)) \cdot \sigma(x) \end{align}$$
$\endgroup$ $\begingroup$Let's say we want to find the derivative of $y=σ(x)=(1+\exp(−x))^{−1}$. So we have:
$$ \begin{align} \frac{dy}{dx} & = (-1)(1 + \exp(-x))^{-2} \frac{d}{dx}(1 + \exp(-x)) \\ \\ & = (-1)(1 + \exp(-x))^{-2}(0 + \frac{d}{dx}\exp(-x)) \\ \\ & = (-1)(1 + \exp(-x))^{-2}(\exp(-x)) \frac{d}{dx}(-x) \\ \\ & = (-1)(1 + \exp(-x))^{-2}(\exp(-x))(-1) \\ \\ & = \frac{\exp(-x)} {(1 + \exp(-x))^2} \\ \\ & = \frac{1 + \exp(-x) -1} {(1 + \exp(-x))^2} \\ \\ & = \frac{1 + \exp(-x)} {(1 + \exp(-x))^2} - \frac{1} {(1 + \exp(-x))^2} \\ \\ & = \sigma(x) - (\sigma(x))^2 \\ \\ & = \sigma(x) \cdot (1 - \sigma(x)) \end{align} $$
$\endgroup$ $\begingroup$By directly differenting:
$$ \sigma^{'} (x)= \frac{1. e^{-x}}{(1+e^{-x})^2} $$
Separately compute, multiply:
$${\sigma(x)}.{(1-\sigma(x))} =\frac{ e^{-x}}{(1+e^{-x}) } . \frac{ 1}{(1+e^{-x}) }$$
The RHSs agree.
EDIT1:
In general a solution of differential equation
$$ \frac{dy}{dx}=y(1-y) $$
can be seen to be
$$\frac{1}{1+c e^{-x}} \rightarrow \frac{1}{1+ e^{-x}} $$
with center point integration constant evaluated at $x=0, y=\frac12;\, c=1. $
$\endgroup$ $\begingroup$No answer yet involves the logarithm. If $\log$ denotes the natural logarithm then by the chain rule$$\frac{d}{dx} \log(\sigma(x)) = \frac{1}{\sigma(x)}\frac{d \sigma(x)}{dx}. $$Furthermore $\log(\sigma(x))=\log(e^x) -\log(1+e^x)=x - \log(1+e^x)$ so that$$\frac{d}{dx} \log(\sigma(x)) = 1 - \frac{e^x}{1+e^x} = 1-\sigma(x).$$Equaling the two displays gives $(d/dx)\sigma(x) = \sigma(x)(1-\sigma(x))$ as desired.
$\endgroup$ $\begingroup$Another approach using the quotient rule is as follows:
let $\sigma=\frac{1}{1+e^-x}$upon rearranging ,
$\sigma=\frac{e^x}{1+e^x}$
using the quotient rule
$\frac{d}{{dx}}\left( {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right) = \frac{{\frac{d}{{dx}}f\left( x \right)g\left( x \right) - f\left( x \right)\frac{d}{{dx}}g\left( x \right)}}{{g^2 \left( x \right)}}$,
let $f(x)=e^x$ and $g(x)=1+e^x$,
we get$\frac{d\sigma}{dx}=\frac{(e^x)(1+e^x)-e^xe^x}{(1+e^x)^2}$
upon rearraning, we get :
$\frac{d\sigma}{dx}=\frac{e^x}{1+e^x}\frac{(1+e^x)-e^x}{1+e^x}$
upon further rearrangement:
$\frac{d\sigma}{dx}=\frac{e^x}{1+e^x}(1-\frac{e^x}{1+e^x})$
$\frac{d\sigma}{dx}=\sigma(1-\sigma)$
$\endgroup$ 1 $\begingroup$$\exp(-x) = \frac{1}{\sigma} -1 $ (By definition). Take the derivative of both sides:
$-\exp(-x)= -\frac{\sigma'}{\sigma^2}$
Add the two to get:$0 = \frac{1}{\sigma} -1 -\frac{\sigma'}{\sigma^2}$
and solve for $\sigma'=\sigma(1-\sigma)$ qed
$\endgroup$ $\begingroup$Using my HP Prime, I differentiated 1/(1+exp(-x)) to get exp(-x)/(1+exp(-x))^2. Factor out 1/(1+exp(-x), which is sigma(x), and the rest is 1-sigma(x). That is proof by calculator. Beware! That machine can become addictive because of the way it amplifies your capability.
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