Logistic Sigmoid and Saturating Growth

Context

The same logistic shape appears as a bounded activation function, a CDF-like response curve, and a simple model of growth that slows as it approaches a capacity. This annex stores the reusable equations so the concept pages can stay navigational.

Formal Statement

The logistic sigmoid is

$$\sigma (x)=\frac{1}{1+e^{-x}},$$

with derivative

$${\sigma }^{\prime }(x)=\sigma (x)(1-\sigma (x)).$$

The logistic growth equation is

$$\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right),$$

where r is the intrinsic growth rate and K is the saturation level.

Derivation / Construction

For the logistic sigmoid, differentiating \sigma(x)=(1+e^{-x})^{-1} gives

$${\sigma }^{\prime }(x)=\frac{e^{-x}}{(1+e^{-x}{)}^2}.$$

Since 1-\sigma(x)=e^{-x}/(1+e^{-x}), this is equivalent to \sigma'(x)=\sigma(x)(1-\sigma(x)).

The logistic growth equation has solution

$$N(t)=\frac{K}{1+A{e}^{-rt}},$$

where A is set by the initial condition. When N \ll K, the factor (1-N/K) is near 1, so growth looks locally exponential; near K, the factor tends to zero.

Implications

• A local exponential fit does not identify the long-run capacity K.
• Saturation claims need a concrete limiting mechanism such as data, compute, energy, adoption, or benchmark headroom.
• Logistic growth is a canonical anchor, not the only possible S-shaped model.

Links

• Sigmoid Functions
• S-Curves and Saturating Growth
• Wikipedia2026 - Sigmoid Function
• LessWrong2026 - All Exponentials are Eventually S-Curves