Sigmoid Function

A sigmoid function (σ) is a mathematical function with an S-shaped curve and a bounded output range (typically between 0 and 1 or -1 and 1).

Although the term “sigmoid function” describes a family of functions, it’s often used as an alias for the logistic function defined by the following equation:

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

The derivative of the logistic sigmoid function is:

$${\sigma }^{\text{'}}(x)=\sigma (x)\cdot (1-\sigma (x))$$

Implementing the sigmoid function in numpy

import numpy as np

def sigmoid(x):
    return 1 / (1 + np.exp(-x))

print(f"σ(0) = {sigmoid(0)}")

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This page references the following sources:

• DeepLearning.AI Deep Learning Specialization (course 1, week 3)
• Google Machine Learning Glossary
• Wikipedia's "Sigmoid function" article
• Wikipedia's "Logistic function" article
• numpy API reference (exp)