What are the mathematical properties of the sigmoid function?

What are the mathematical properties of the sigmoid function?

The sigmoid function is defined as ( S(x) = \frac{1}{1 + e^{-x}} ). It has several key properties.

What are those key properties?

Some key properties include:

1. Range: The output is between 0 and 1.
2. Derivative: The derivative is ( S'(x) = S(x)(1 - S(x)) ).
3. Monotonicity: It is a monotonically increasing function.

How does the sigmoid function behave as x approaches infinity?

As ( x ) approaches infinity, the sigmoid function approaches 1. Conversely, as ( x ) approaches negative infinity, it approaches 0.

Can you explain its significance in machine learning?

Certainly! The sigmoid function is often used as an activation function in neural networks, helping to model probabilities.