Sigmoid function
The logistic curve y = 1/(1 + e^(−kx)): growth that starts slow, speeds up, and levels off at 1.
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Formulas and key values
- Cartesian equation
- Derivative
- Steepest slope · at the inflection point (0, 1/2)In the graph = 0.25
- Asymptote
k sets the steepness; the curve rises from 0 to 1 and is centred at x = 0.
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Start the video01 · Definition
What is the Sigmoid function?
Pierre-François Verhulst introduced the logistic curve in the late 1830s to model a population that grows in proportion to its size until resources run short. Its differential equation y′ = k·y(1 − y) says exactly that: growth is fast when y is small and stops as y approaches the ceiling 1.
The same S-shape reappears wherever a quantity switches smoothly between two levels: as the activation function of early neural networks, in logistic regression, in dose–response curves and in the adoption of new technology. Its steepest point is the inflection (0, 1/2), where the slope is k/4.
02 · Properties
Key properties
- Horizontal asymptotes y = 0 and y = 1.
- Inflection point (0, 1/2), where the slope k/4 is the largest.
- Derivative y′ = k·y(1 − y).
- Point symmetry: σ(−x) = 1 − σ(x).
- Inverse function: the logit, x = (1/k)·ln(y/(1 − y)).
03 · Questions
Sigmoid function: common questions
- What is the equation of the Sigmoid function?
- Cartesian equation: y = 1/(1 + e^(-kx)). k sets the steepness; the curve rises from 0 to 1 and is centred at x = 0.
04 · Keep exploring