[Math] Derivative of Logistic Function

Derivative (도함수) of Logistic Function (Sigmoid라고도 불림)

$y=\sigma(x)=\dfrac{1}{1+e^{-x}}$ 를 미분하면 다음과 같음.

$$\frac{d}{dx}\sigma(x)= \sigma(x)(1-\sigma(x))$$

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graph

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유도

유도는 다음과 같음.

$$\begin{aligned}\frac{d}{dx}\sigma(x)&= \dfrac{0\times(1+e^{-1}) - 1\times(-e^{-1})}{(1+e^{-x})^2}\\&=\dfrac{e^{-x}}{(1+e^{-x})^2}\\ &= \frac{1}{(1+e^{-x})}\frac{e^{-x}}{(1+e^{-x})}\\ &= \frac{1}{(1+e^{-x})}\left( 1-\dfrac{1}{(1+e^{-x})}\right) \\ &= \sigma(x) (1-\sigma(x)) \end{aligned}$$

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참고자료

2024.02.28 - [.../Math] - [Math] The Key Rules of Differentiation

[Math] The Key Rules of Differentiation