LLM Primer

Chapter 6

Calculus

Derivatives, gradients, Jacobians, chain rule, computation graphs, and backpropagation.

What this chapter does

Calculus is the language of local change. This chapter starts with slope, then builds toward gradients, Jacobians, computation graphs, and backpropagation as organized chain rule.

Lessons

Read these in order.

Start with the chapter introduction, then move through the topic lessons. The order is chosen so each page can reuse ideas from the pages before it.

  1. 01
    Introduction

    Why calculus is the language of change in machine learning.

  2. 02
    Change and Slope

    Slope as change in output divided by change in input.

  3. 03
    Derivatives

    Derivatives as local sensitivity of a function.

  4. 04
    Partial Derivatives

    Changing one input while holding the others fixed.

  5. 05
    Gradients

    Collecting partial derivatives into the vector used for learning.

  6. 06
    Directional Derivatives

    Using dot products to measure change along a chosen direction.

  7. 07
    Jacobians and Hessians

    Vector-output sensitivity and curvature at a working level.

  8. 08
    Chain Rule

    Multiplying local rates through composed functions.

  9. 09
    Computation Graphs

    Dependency maps for forward values and backward sensitivities.

  10. 10
    Backpropagation

    Chain rule applied backward through a computation graph.

Review and practice

Close the chapter deliberately.

Use the conclusion and revision notes before the chapter exercises. Hints and solutions are collected here, while lesson-level exercises reveal their own help inline.

Conclusion

What Chapter 6 accomplished and how calculus prepares probability.

Summary and Revision Notes

A compact review of derivatives, gradients, chain rule, and backpropagation.

Exercises

Chapter-level practice for local change, gradients, graphs, and backpropagation.

Hints

Low-spoiler nudges for the Chapter 6 exercises.

Solutions

Explained solutions for the Chapter 6 exercises.

Before moving on

  • Follow how loss changes with parameters.
  • Understand backpropagation as repeated chain rule.
  • Read gradients as vectors of local sensitivity.
  • Separate local derivative information from a full training step.

Where this leads

  • Optimization
  • Deep Learning

Chapter progress