# Sigmoid Calculator

URL: https://speedycalculator.com/calculators/math/sigmoid/
Last updated: 2026-10-30

> The sigmoid calculator evaluates the logistic function σ(x) = 1/(1 + e⁻ˣ), its derivative and its inverse (the logit).

## Inputs

- Finite real input x
- Probability for the logit (0 to 1)

## Results

- σ(x)
- Derivative σ′(x)
- logit(p) = ln(p/(1 − p))
- tanh(x/2) = 2σ(x) − 1

## Formula

```
σ(x) = 1 / (1 + e^(−x)),  σ′(x) = σ(x)(1 − σ(x)),  logit(p) = ln(p / (1 − p))
```

## Explanation

The sigmoid squashes any real number into the range 0 to 1, so it is used for probabilities in logistic regression, as an activation function in neural networks and in growth models that level off.

### How to use the Sigmoid Calculator

- **Finite real input x**
- **Probability for the logit (0 to 1)**

Results update as you type, and everything is calculated in your browser.

### How it works

σ(0) = 0.5; large positive x gives values near 1, large negative x values near 0. The inverse, the logit, turns a probability back into log-odds.

### Inputs explained

FieldWhat to enterFinite real input xA numberProbability for the logit (0 to 1)A number

### What each result means

- **σ(x)** = 1÷(1 + exp(-[Finite real input x]))
- **Derivative σ′(x)** = 1÷(1 + exp(-[Finite real input x]))×(1 - 1÷(1 + exp(-[Finite real input x])))
- **logit(p) = ln(p/(1 − p))** = ln([Probability for the logit (0 to 1)]÷(1 - [Probability for the logit (0 to 1)]))
- **tanh(x/2) = 2σ(x) − 1** = 2÷(1 + exp(-[Finite real input x])) - 1

### Example values

How the main result changes when **Finite real input x** changes and everything else stays at the example values above:

Finite real input xσ(x)0.750.67921.130.75491.50.81761.880.8672.250.904730.9526

## Worked example

Inputs — Finite real input x: 1.5; Probability for the logit (0 to 1): 0.8.

Results — **σ(x) = 0.81757448**; Derivative σ′(x) = 0.14914645; logit(p) = ln(p/(1 − p)) = 1.38629436; tanh(x/2) = 2σ(x) − 1 = 0.63514895.

## Key terms

- **Logistic function**: σ(x) = 1/(1 + e⁻ˣ).
- **Logit**: ln(p/(1 − p)).

## FAQ

**What is σ(0)?**
0.5.

**Why is the sigmoid used in neural networks?**
It is smooth and differentiable and maps outputs to (0, 1); modern networks often prefer ReLU in hidden layers.

**How is it related to tanh?**
tanh(x) = 2σ(2x) − 1.

**What is the logit?**
The inverse of the sigmoid: the log-odds of a probability.


## Sources

- [Wolfram MathWorld – Sigmoid function](https://mathworld.wolfram.com/SigmoidFunction.html)
- [OpenStax – Calculus Volume 1](https://openstax.org/details/books/calculus-volume-1)