Editing Total derivative (section)

===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 <math>L(t,x_1,\dots,x_n)</math> is a function of time <math>t</math> and <math>n</math> variables <math>x_i</math> which themselves depend on time. Then, the time derivative of <math>L</math> is
:<math>\frac{dL}{dt} = \frac{d}{dt} L \bigl(t, x_1(t), \ldots, x_n(t)\bigr).</math>

The chain rule expresses this derivative in terms of the partial derivatives of <math>L</math> and the time derivatives of the functions <math>x_i</math>:
:<math>\frac{dL}{dt}
= \frac{\partial L}{\partial t} + \sum_{i=1}^n \frac{\partial L}{\partial x_i}\frac{dx_i}{dt}
= \biggl(\frac{\partial}{\partial t} + \sum_{i=1}^n \frac{dx_i}{dt}\frac{\partial}{\partial x_i}\biggr)(L).</math>

This expression is often used in [[physics]] for a [[gauge transformation]] of the [[Lagrangian mechanics|Lagrangian]], as two Lagrangians that differ only by the total time derivative of a function of time and the <math>n</math> [[generalized coordinates]] lead to the same equations of motion. An interesting example concerns the resolution of causality concerning the [[Wheeler–Feynman absorber theory#Resolution of causality issue|Wheeler–Feynman time-symmetric theory]].  The operator in brackets (in the final expression above) is also called the total derivative operator (with respect to <math>t</math>).

For example, the total derivative of <math>f(x(t),y(t))</math> is

:<math>\frac{df}{dt} = { \partial f \over \partial x}{dx \over dt} + {\partial f \over \partial y}{dy \over dt }.</math>

Here there is no <math>\partial f / \partial t</math> term since <math>f</math> itself does not depend on the independent variable <math>t</math> directly.