The chain rule extends from two-function compositions to compositions of three (or more) functions by introducing an intermediate variable for each nested layer and multiplying the derivatives of each layer with respect to the next-outer variable in sequence. Formally, for y = f(g(h(θ))), the derivative dy/dθ is computed as dy/dx · dx/dw · dw/dθ, where x and w are auxiliary variables representing the intermediate compositions. This belongs to single-variable differential calculus and generalizes the two-function chain rule, requiring correct identification of the outermost, middle, and innermost functions in a nested composition.
The chain rule extends from two-function compositions to compositions of three (or more) functions by introducing an intermediate variable for each nested layer and multiplying the derivatives of eac…