sigmoid function mathematics Ask the Chatbot a Question More Actions Print Cite verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style MLA APA Chicago Manual of Style Copy Citation Share Share Share to social media Facebook X URL https://www.britannica.com/science/sigmoid-function Feedback Feedback Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login). Feedback Type Select a type (Required) Factual Correction Spelling/Grammar Correction Link Correction Additional Information Other Your Feedback Submit Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article.

　　External Websites Ask the Chatbot a Question Also known as: standard logistic function Written by L. Sue Baugh L. Sue Baugh is a writer and editor who works in the educational, business, and science/medical fields. L. Sue Baugh Fact-checked by The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors. The Editors of Encyclopaedia Britannica Last Updated: Dec 6, 2024 • Article History Table of Contents Table of Contents Ask the Chatbot a Question Also called: standard logistic function (Show more) { "@context": "https://schema.org", "@type": "FAQPage", "mainEntity": [ { "@type" : "Question", "name" : " What is the sigmoid function? ", "acceptedAnswer" : { "@type" : "Answer", "text" : "The sigmoid function, also known as the standard logistic function, is a mathematical function that graphs as an S-shaped curve. It is represented by the equation u03C3(x) = 1/(1 + eu2212x)." } } , { "@type" : "Question", "name" : " How does the sigmoid function behave for large values of x? ", "acceptedAnswer" : { "@type" : "Answer", "text" : "For large negative values of x, u03C3(x) approaches 0, and for large positive values of x, u03C3(x) approaches 1." } } , { "@type" : "Question", "name" : " What role did the sigmoid function play in neural networks? ", "acceptedAnswer" : { "@type" : "Answer", "text" : "The sigmoid function was used as an activation function in early neural networks, which was useful for binary classification and handling nonlinear relationships among data." } } , { "@type" : "Question", "name" : " Why is the sigmoid function less used in modern neural networks? ", "acceptedAnswer" : { "@type" : "Answer", "text" : "In modern neural networks, the sigmoid function has been replaced by specially designed activation functions that are faster and more economical, though often derived from the classic sigmoid function." } } ] } Top Questions What is the sigmoid function? The sigmoid function, also known as the standard logistic function, is a mathematical function that graphs as an S-shaped curve. It is represented by the equation σ(x) = 1/(1 + e−x).

　　How does the sigmoid function behave for large values of x? For large negative values of x, σ(x) approaches 0, and for large positive values of x, σ(x) approaches 1.

　　What role did the sigmoid function play in neural networks? The sigmoid function was used as an activation function in early neural networks, which was useful for binary classification and handling nonlinear relationships among data.

　　Why is the sigmoid function less used in modern neural networks? In modern neural networks, the sigmoid function has been replaced by specially designed activation functions that are faster and more economical, though often derived from the classic sigmoid function.

　　sigmoid function, mathematical function that graphs as a distinctive S-shaped curve. The mathematical representation of the sigmoid function is an exponential equation of the formσ(x) = 1/(1 + e−x),where e is the constant that is the base of the natural logarithm function.

　　Although there are many S-shaped, sigmoidlike curves, it is the standard form of the logistic function that is referred to as the “sigmoid.” The logistic function was first derived by Belgian mathematician Pierre-François Verhulst in the mid-1830s to describe population growth.

　　The sigmoid function has the behavior that for large negative values of x, σ(x) approaches 0, and for large positive values of x, σ(x) approaches 1. The derivative of the sigmoid function isd(σ(x))e/dx = e−x/(1 + ex)2.

　　The sigmoid function played a key part in the evolution of neural networks and machine learning. A neural network is a computer network that operates similarly to the way neurons operate in the brain. A neuron in a neural network receives input from other neurons, and that input is sent into an activation function that determines the output.

　　Often the activation function was a sigmoid. The function’s outputs of 0 and 1 were useful in problems with binary classification. Its nonlinearity property was required to make complex decisions in networks in which there were nonlinear relationships among data. Because of these properties, the sigmoid function became an essential component in early neural networks, and it was therefore often referred to as the “sigmoid” or “sigmoid unit.”

　　In modern neural networks, the traditional sigmoid function σ(x) has often been replaced by specially designed activation functions that are faster and more economical to use. Nevertheless, these new activation functions are usually created by modifying the classic sigmoid function σ(x).