linguistic:translation, structural:expansion
Example: Differentiation with indirect dependencies

While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose L (t, x 1, …, x n) {\displaystyle L(t,x_{1},\dots,x_{n})} is a function of time t {\displaystyle t} and n {\displaystyle n} variables x i {\displaystyle x_{i}} which themselves depend on time. Then, the time derivative of L {\displaystyle L} is

The chain rule expresses this derivative in terms of the partial derivatives of L {\displaystyle L} and the time derivatives of the functions x i {\displaystyle x_{i}}:

This expression is often used in physics for a gauge transformation of the Lagrangian, as two Lagrangians that differ only by the total time derivative of a function of time and the n {\displaystyle n} generalized coordinates lead to the same equations of motion. An interesting example concerns the resolution of causality concerning the Wheeler–Feynman time-symmetric theory. The operator in brackets (in the final expression above) is also called the total derivative operator (with respect to t {\displaystyle t}).

For example, the total derivative of f (x (t), y (t)) {\displaystyle f(x(t),y(t))} is

Here there is no ∂ f / ∂ t {\displaystyle \partial f/\partial t} term since f {\displaystyle f} itself does not depend on the independent variable t {\displaystyle t} directly.