- 1 The Maths behind Neural Networks
- 2 The Simplest Neural Network: Perceptron using Vectors and Dot Products
- Vectors
- Matrices: a way to represent vectors (and tensors)
- Dot Product
- Vector Space and Hyperplane
- 3 Perceptron Training via Feature Vectors and Hyperplane Split
- The Intuition: Nudging the Vector
- The Update Rules
- The Formal Algorithm
- 4 Gradient Descent
- Neural Network as a Chain of Functions
- Forward Pass and the Cost Function
- Gradient Descent for Scalar Functions: a worked example
- Applying the Chain Rule
- Gradient Descent for a Two-Layer Neural Network
- 5 Backpropagation with Scalar Calculus
- How Backpropagation Works
- Writing it out as the Chain Rule
- The Backpropagation Algorithm Step by Step
- 6 Backpropagation with Matrix Calculus
- Gradient Vector
- Jacobian Matrix and the Chain Rule with Matrices
- VJP (Vector-Jacobian Product) and JVP
- Hadamard Product
- Backpropagation Derivation for a 2-Layer Network
- Backpropagation Equations in NumPy
- 7 Backpropagation with Softmax and Cross-Entropy
- Cross-Entropy Loss w.r.t. Weights in the Last Layer
- Derivative of Loss w.r.t. Weights in Inner Layers
- Implementation in Python
- 8 Neural Network Implementation
- A Two-Layer Neural Network in Python
The Mathematical Intuition Behind Deep Learning
From the Dot Product to Multivariate Calculus and the Jacobian, with Python
Stop treating neural networks as magic. Starting from a single dot product, this book derives every step to Backpropagation with Matrix Calculus, then turns the math into working Python. No frameworks, no skipped steps. Written for developers who want to know why the equations work, not just how to call them.
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About
About the Book
Most deep learning resources either jump straight to model.fit() or bury you in notation. This book takes the path in between: every idea is built from intuition, derived step by step, and then written as code you can run.
What you will learn:
- See the geometry: how a Perceptron separates data using vectors, dot products and hyperplanes.
- Watch learning happen: how Gradient Descent follows the slope of a loss function, and why the learning rate matters.
- Derive Backpropagation yourself: first with the Chain Rule on a tiny network, then generalised with Matrix Calculus.
- Build a network from scratch: implement forward and backward passes in plain Python and NumPy, with no framework magic.
- Understand the modern pieces: Softmax, Cross-Entropy Loss, and the matrix operations behind today's large language models.
Who it's for: software engineers, students and ML practitioners who know basic Python and school-level algebra. The calculus you need is explained when you need it.
Why this book: it grew out of my "Explained as Simple as Possible" articles on Medium, which have been viewed over 100,000 times. Readers kept asking for the full derivations in one place, with nothing skipped. This is that book.
Free updates for life: new chapters and notebooks are added as the book grows.
Author
About the Author
Alex Punnen
Alex Punnen is a software architect with over 20 years of experience building large-scale distributed systems, from early middleware to cloud-native platforms. He is a senior technical leader at a global telecom company, where he leads AI-native software engineering across multiple teams.
He writes about AI, LLMs and software engineering on Medium, where his "Explained as Simple as Possible" articles on transformers, MCP and agentic AI have been featured in Towards AI and read by a wide developer audience.
He learns by deriving things from first principles, and this book is the result of doing exactly that for deep learning.
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Table of Contents
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