You have learned how to compute derivatives and gradients for functions. Machine learning models, particularly neural networks, often involve complex structures where functions are nested within each other. Calculating the gradient of the overall model's error with respect to its internal parameters requires a systematic way to handle these nested dependencies.

This chapter introduces the chain rule, a core calculus concept for differentiating composite functions. We will start with the single-variable chain rule and extend it to multivariable functions. You will see how neural networks can be mathematically represented as compositions of functions and understand how the chain rule forms the basis of the backpropagation algorithm. Backpropagation is the standard method for efficiently calculating the gradients needed to update the weights and biases during neural network training. We will also look at computational graphs to help visualize the flow of computation and gradient calculation. Upon completing this chapter, you will grasp the mechanics behind how gradients are computed in deep learning models.