Sigmoid Function: Derivative and Working Mechanism

Jp Patel 21 Mar, 2024
6 min read


In deep learning, the activation functions are one of the essential parameters in training and building a deep learning model that makes accurate predictions. Choosing the best appropriate activation function can help one get better results with even reduced data quality; hence, activation functions should be decided according to their characteristics and behavior on the fed data. The sigmoid function derivative, for instance, is a popular choice for activation in certain layers of neural networks due to its ability to squash the output between 0 and 1, which is useful for binary classification tasks.

This article will discuss one of the most famous and used accusation functions, the sigmoid. We will calculate its derivative to understand the core intuition and working mechanism behind it, and we will also discuss the applications and advantages of the activation function.

This article was published as a part of the Data Science Blogathon.

Sigmoid Function

The sigmoid function is one of the most used activation functions in machine learning and deep learning. It can be used in the hidden layers, which take the previous layer’s output and bring the input values between 0 and 1. Now while working with neural networks, it is necessary to calculate the derivate of the activation function.

Visual representation of data being fed into the sigmoid function

The formula of the sigmoid activation function is:

F(x) = σ(x) = 1 ⁄ (1 + e-x)

The graph of the sigmoid function looks like an S curve, where the part of the function is continuous and differential at any point in its area.

The sigmoid function, also known as the squashing function, takes the input from the previously hidden layer and squeezes it between 0 and 1. So a value fed to the function will always return a value between 0 and 1, no matter how big or small the deal is provided.

Graphically, the sigmoid function looks like this,

Graph of the Sigmoid function looking like an 'S'

Also Read: How to Understand Sigmoid Function in Artificial Neural Networks?

Derivative of the Sigmoid Function

In neural networks, the weights and biases are assigned randomly in the initial stages, and the weights and biases get updated during the backpropagation in the network. Here while performing the backpropagation, the derivatives are calculated. Also, the derivative of the activation function is calculated. The sigmoid function is the only activation function which haves its own function in its derivative.

Let’s try to derive the same.

The formula of the Sigmoid Function: 

σ(x) = 1/1 + e-x

Step 1: Derivation Concerning x on Both Sides.

σ(x) = d/dx {1/(1 + e-x)}

Step 2: Applying the Reciprocating Rule.

σ(x)’ = d/dx ( (1/1 + e-x)-1)

σ(x)’ = – 1/ (1 + e-x)2 d/dx (1+e-x)

σ(x)’ = – 1/ (1 + e-x)2 d/dx (e-x)

σ(x)’ = – e-x / ( 1+ e-x)2 d/dx (-x)

σ(x)’ = e-x / ( 1 + e-x )

The above equation is known as the derivative of the sigmoid function.

Modifying the Equation for a More Generalized Form

σ(x)’ = e-x / ( 1 + e-x) ( 1+ e-x)

σ(x)’ = 1/ 1 + e-x × e-x/1+e-x

σ(x)’ = σ(x) × e-x/1+e-x

σ(x)’ = σ(x) × ( 1 – 1/1+e-x)

σ(x)’ = σ(x) ( 1- σ(x) )

The above equation is known as the generalized form of the sigmoid function.

Graph of the Sigmoid function and its derivative

Code to Implement Sigmoid Function

One must write the following code to implement the sigmoid function in Python. The below required the value of x to be pre-defined to get the sigmoid deal out of it.

Python Code:

Suppose the input weights are fed to the layers, and the consequences and biases are passed into the next layer. If the final output layer has a sigmoid function, it will be applied to the output, and the final result will be displayed.

For example,

Let’s suppose the output from the hidden layer is 1; then, the value of x would be 1.

Final Output:

= 1 / 1 + e-x

= 1 / 1 + e-1

= 1/ 1 + 0.367

= 1 / 1.367

= 0.7315

As we can see here, the output from the previously hidden layer was 1, and the function made it 0.7315, where it is visible that the Sigmoid Function is a Squeezing Function.

Applications of Sigmoid Function

1. Binary Classification Problems:

We can use the sigmoid function in binary classification problems as it returns the output between 0 and 1.

2. Probabilistic Models:

We can use the sigmoid function when we are required to work on a probabilistic model as it can be used to calculate the probability of a given class between 0 and 1.

3. Image Datasets and Neural Networks:

The sigmoid function can be used for neural networks on image datasets for performing tasks like image segmentations, classifications, etc.

Limitations of Sigmoid Function

Vanishing and exploding gradients of the Sigmoid function and its derivative

1. Vanishing Gradient Problem:

One of the significant issues with the sigmoid is the lack of weight updating. Therefore, the function sometimes returns small values as outputs, making no changes in the weights and biases that cause the vanishing gradient problem.

2. Exploding Gradient Problem:

It also sometimes happens that the sigmoid returns very large values as output, resulting in an exploding gradient problem.

3. A Squeezing Function:

It is sometimes noted that as the sigmoid is a squeezing function, it makes the output results between 0 and 1, thereby hiding the essence or the knowledge between much higher and much smaller numbers and making the model less accurate.

Linear vs. non-Linearly Separable Problems

  • Linear problems can be separated by a single straight line/hyperplane. These are relatively straightforward and can be solved by linear models like logistic regression or linear SVMs.
  • Non-linearly separable problems have data points from different classes intricately intertwined, requiring more complex decision boundaries.
  • Neural networks excel at non-linearly separable problems due to their ability to learn complex non-linear functions through:
  • Choice of activation function is crucial:
    1. Sigmoid function is good for binary classification but suffers from vanishing/exploding gradient problems
    2. Advanced activations like ReLU mitigate vanishing gradients for better training
  • Limitations of sigmoid activation:
    1. Vanishing gradient problem
    2. Exploding gradient problem
    3. Restricted 0-1 output range limits capturing patterns in extreme values
  • Neural network architectures and components like activation functions, regularization enable modeling of complex non-linear patterns across domains like:
    1. Image recognition
    2. Natural language processing
    3. Speech recognition
    4. Reinforcement learning


In this article, we discussed the sigmoid function and its derivative, its working mechanism, and the core intuition behind the same with its applications associated with advantages and disadvantages. Knowing these key concepts will help one better understand the mathematics behind the function and will help one answer any related interview questions efficiently.

Key Takeaway

  • Sigmoid function is a squeezing function that results from the output between 0 and 1.
  • The Sigmoid can be used efficiently for binary classification problems, as it returns the output between 0 and 1.
  • The function sometimes returns much larger or smaller values, resulting in vanishing or exploding gradient problems.

Frequently Asked Questions?

1. Why is backpropagation accessible to the sigmoid function?

Since it is the only activation function that appears in the derivative of itself, it helps neural networks better perform the backpropagation algorithm, as the gradient descent is used for updating weights and biases of the model.

2. Why Sigmoid Activation function is squeezing function?

As the activation function squeezes the input values fed to the hidden layers, the function returns the output between 0 and 1. So no matter how positive or negative numbers are provided to the layer, this function squeezes it between 0 and 1.

3. What is the main issue with the sigmoid function during backpropagation?

The main issue related to the activation function is when the gradient descent algorithm calculates the new weights and biases; if these values are minimal, then the updates of the consequences and preferences will also be meager and hence, which results in a vanishing gradient problem, where the model will not lean anything.

4. What is the range of the sigmoid derivative?

The range of the sigmoid derivative is between 0 and 0, inclusive, for all real values of x.

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Jp Patel 21 Mar, 2024

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