derivative of the sigmoid function

Daniel Parker

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18-03-2025

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The sigmoid function, also known as the logistic function, is a crucial element in various fields, particularly in machine learning and deep learning. Understanding its derivative is essential for employing optimization algorithms like gradient descent in neural networks. This article provides a comprehensive explanation of the sigmoid function and a detailed derivation of its derivative.

What is the Sigmoid Function?

The sigmoid function is a mathematical function having a characteristic "S"-shaped curve. It maps any input value (usually a real number) to an output value between 0 and 1. This makes it particularly useful for representing probabilities or activation levels in neural networks. The function is defined as:

σ(x) = 1 / (1 + e^(-x))

Where:

• σ(x) represents the sigmoid function of x.
• e is Euler's number (approximately 2.71828).
• x is the input value.

The sigmoid function's output approaches 0 as x approaches negative infinity and approaches 1 as x approaches positive infinity. This bounded output range is a key property exploited in many applications.

Why is the Derivative Important?

In machine learning, especially within neural networks, we often need to adjust the weights and biases of the network to minimize the error. Gradient descent is a common optimization algorithm that uses the derivative of the activation function (like the sigmoid) to determine the direction and magnitude of these adjustments. Without the derivative, we wouldn't be able to effectively train these networks.

Deriving the Derivative of the Sigmoid Function

To find the derivative of the sigmoid function, we'll use the chain rule of calculus. The chain rule states that the derivative of a composite function is the derivative of the outer function (with the inside function left alone) times the derivative of the inside function.

Let's break down the derivation step-by-step:

1. Rewrite the function: We can rewrite the sigmoid function as:

   σ(x) = (1 + e^(-x))^(-1)

2. Apply the chain rule: The outer function is u^(-1), and the inner function is u = 1 + e^(-x). The derivative of the outer function is -u^(-2), and the derivative of the inner function is -e^(-x).

3. Substitute and simplify: Applying the chain rule, we get:

   dσ(x)/dx = - (1 + e^(-x))^(-2) * (-e^(-x))

4. Simplify further: We can simplify this expression:

   dσ(x)/dx = e^(-x) / (1 + e^(-x))^2

5. An alternative, elegant form: We can express this in a more elegant and computationally efficient way using the original sigmoid function:

   dσ(x)/dx = σ(x) * (1 - σ(x))

This final form is particularly useful because it only requires a single computation of the sigmoid function itself, making it computationally efficient for backpropagation in neural networks.

Applications of the Sigmoid Derivative

The derivative of the sigmoid function plays a crucial role in:

• Backpropagation: During training of neural networks, the derivative is used to calculate the gradient of the loss function with respect to the network's weights and biases. This gradient guides the weight updates in gradient descent.
• Gradient Descent Optimization: The derivative dictates the direction and size of parameter updates during training.
• Probability Estimation: In logistic regression, the sigmoid function outputs probabilities, and its derivative helps in estimating the uncertainty associated with these probabilities.

Conclusion

Understanding the derivative of the sigmoid function is paramount for anyone working with neural networks or machine learning models that utilize this activation function. Its derivation, while initially seeming complex, simplifies to a remarkably concise and computationally efficient form that is fundamental to the training and optimization processes within these models. The σ(x) * (1 - σ(x)) form is highly recommended for practical applications due to its efficiency.