Conceptual
Browse Instruct Curate
Login

The Chain Rule for Composite and Multivariable Functions

The chain rule states that the derivative of a composition f(g(x)) is f'(g(x))·g'(x), and its multivariable form sums the contributions of every path by which one variable influences another: for z depending on x through several intermediates, dz/dx is the sum over intermediates of (partial z / partial u_i)(du_i / dx). Learners compute derivatives of nested expressions, distinguish partial from total derivatives, and read a chain of multiplied local derivatives as a single end-to-end sensitivity.