This video provides a formal, calculus-based explanation of the backpropagation algorithm, building on an intuitive understanding. It meticulously applies the chain rule to derive the sensitivity of a neural network's cost function to its weights and biases, first with a simple single-neuron-per-layer network and then extending to multi-neuron layers. The core message emphasizes how these derivatives form the gradient used to iteratively minimize the network's cost, highlighting backpropagation as the workhorse of deep learning.