Step by step solution to "What is the derivative of sigmoid function: σ(x)=11+e−x"

Madeline Lott

Answered question

2021-12-11

What is the derivative of sigmoid function:

σ (x) = \frac{1}{1 + e^{- x}}

Answer & Explanation

Mary Herrera

Beginner2021-12-12Added 37 answers

The derivate is:
$\frac{d}{d x} σ (x) = σ (x) (1 - σ (x))$
Then,
$\frac{d}{d x} σ (x) = \frac{d}{d x} [\frac{1}{1 + e^{x}}]$ =
= $\frac{d}{d x} {(1 + e^{x})}^{- 1}$ =
= $- {(1 + e^{- x})}^{- 2} (- e^{- x})$ =
= $\frac{e^{- x}}{{(1 + e^{- x})}^{2}}$ =
= $\frac{1}{1 + e^{- x}} \times \frac{e^{- x}}{1 + e^{- x}}$ =
= $\frac{1}{1 + e^{- x}} \times \frac{(1 + e^{- x}) - 1}{1 + e^{- x}}$ =
= $\frac{1}{1 + e^{- x}} \times (\frac{1 + e^{- x}}{1 + e^{- x}} - \frac{1}{1 + e^{- x}})$ =
= $\frac{1}{1 + e^{- x}} \times (1 - \frac{1}{1 + e^{- x}})$ =
$= σ (x) \times (1 - σ (x))$

eninsala06

Beginner2021-12-13Added 37 answers

f (x) = \frac{1}{σ (x)} = 1 + e^{- x}

f^{'} (x) = \frac{d}{d x} (\frac{1}{σ (x)}) = - \frac{σ^{'} (x)}{σ {(x)}^{2}}

,
Thus,

f^{'} (x) = \frac{d}{d x} (1 + e^{- x}) = - e^{- x} = 1 - f (x) = 1 - \frac{1}{σ (x)} = - \frac{σ (x) - 1}{σ (x)}

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