In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments.
Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one.
Conceptually, the definition of the total derivative expresses the idea that
This can be made precise by quantifying the error in the linear approximation determined by
is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain.
The converse does not hold: it can happen that all of the partial derivatives of
is the linear transformation corresponding to the Jacobian matrix of partial derivatives at that point.
[2] When the function under consideration is real-valued, the total derivative can be recast using differential forms.
may be written in terms of its Jacobian matrix, which in this instance is a row matrix: The linear approximation property of the total derivative implies that if is a small vector (where the
are infinitesimal increments in the coordinate directions, then In fact, the notion of the infinitesimal, which is merely symbolic here, can be equipped with extensive mathematical structure.
Techniques, such as the theory of differential forms, effectively give analytical and algebraic descriptions of objects like infinitesimal increments,
may be inscribed as a linear functional on the vector space
The chain rule has a particularly elegant statement in terms of total derivatives.
are identified with their Jacobian matrices, then the composite on the right-hand side is simply matrix multiplication.
This is enormously useful in applications, as it makes it possible to account for essentially arbitrary dependencies among the arguments of a composite function.
, then the behavior of f may be understood in terms of its partial derivatives in the x and y directions.
In this case, we are actually interested in the behavior of the composite function
However, the chain rule for the total derivative takes such dependencies into account.
Then, the chain rule says By expressing the total derivative using Jacobian matrices, this becomes: Suppressing the evaluation at
for legibility, we may also write this as This gives a straightforward formula for the derivative of
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 the partial derivative assumes that y is fixed.
Instead of immediately substituting for y in terms of x, however, we can also use the chain rule as above: While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique.
: 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
generalized coordinates lead to the same equations of motion.
A total differential equation is a differential equation expressed in terms of total derivatives.
Since the exterior derivative is coordinate-free, in a sense that can be given a technical meaning, such equations are intrinsic and geometric.
In economics, it is common for the total derivative to arise in the context of a system of equations.[1]: pp.
217–220 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.
of p with respect to r, for example, gives the sign and magnitude of the reaction of the market price to the exogenous variable r. In the indicated system, there are a total of six possible total derivatives, also known in this context as comparative static derivatives: dp / dr, dp / dw, dp / dI, dq / dr, dq / dw, and dq / dI.
The total derivatives are found by totally differentiating the system of equations, dividing through by, say dr, treating dq / dr and dp / dr as the unknowns, setting dI = dw = 0, and solving the two totally differentiated equations simultaneously, typically by using Cramer's rule.