In the mathematical field of differential calculus, a total derivative or full derivative of a function
Contents
- Differentiation with indirect dependencies
- The total derivative via differentials
- The total derivative as a linear map
- Total differential equation
- Application to equation systems
- References
The total derivative of a function is different from its corresponding partial derivative (
which simplifies to
Consider multiplying both sides of the equation by the differential
The result is the differential change
"Total derivative" is sometimes also used as a synonym for the material derivative,
Differentiation with indirect dependencies
Suppose that f is a function of two variables, x and y. Normally these variables are assumed to be independent. However, in some situations they may be dependent on each other. For example y could be a function of x, constraining the domain of f to a curve in
For example, suppose
The rate of change of f with respect to x is usually the partial derivative of f with respect to x; in this case,
However, if y depends on x, the partial derivative does not give the true rate of change of f as x changes because it holds y fixed.
Suppose we are constrained to the line
then
In that case, the total derivative of f with respect to x is
Instead of immediately substituting for y in terms of x, this can be found equivalently using the chain rule:
Notice that this is not equal to the partial derivative:
While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose M(t, p1, ..., pn) is a function of time t and n variables
The chain rule for differentiating a function of several variables implies that
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 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) is also called the total derivative operator (with respect to t).
For example, the total derivative of f(x(t), y(t)) is
Here there is no ∂f / ∂t term since f itself does not depend on the independent variable t directly.
The total derivative via differentials
Differentials provide a simple way to understand the total derivative. For instance, suppose
This expression is often interpreted heuristically as a relation between infinitesimals. However, if the variables t and
Dividing through by dt gives the total derivative dM / dt.
The total derivative as a linear map
Let
The linear map
Note that f is differentiable if and only if each of its components
Total differential equation
A total differential equation is a differential equation expressed in terms of total derivatives. Since the exterior derivative is a natural operator, in a sense that can be given a technical meaning, such equations are intrinsic and geometric.
Application to equation systems
In economics, it is common for the total derivative to arise in the context of a system of equations. For example, a simple supply-demand system might specify the quantity q of a product demanded as a function D of its price p and consumers' income I, the latter being an exogenous variable, and might specify the quantity supplied by producers as a function S of its price and two exogenous resource cost variables r and w. The resulting system of equations,
determines the market equilibrium values of the variables p and q. The total derivative of, for example, p with respect to r,