Total derivative

# Total derivative

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In the mathematical field
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

of differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

, the term total derivative has a number of closely related meanings.

• The total derivative (full derivative) of a function , of several variables, e.g., , , , etc., with respect to one of its input variables, e.g., , is different from the partial derivative ()
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

. Calculation of the total derivative of with respect to does not assume that the other arguments are constant while varies; instead, it allows the other arguments to depend on . The total derivative adds in these indirect dependencies to find the overall dependency of on . For example, the total derivative of with respect to is

Consider multiplying both sides of the equation by the differential :

The result will be the differential change in the function . Because depends on , some of that change will be due to the partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

of with respect to . However, some of that change will also be due to the partial derivatives of with respect to the variables and . So, the differential is applied to the total derivatives of and to find differentials and , which can then be used to find the contribution to .

• It refers to a differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

such as

which computes the total derivative of a function (with respect to x in this case).

• It refers to the (total) differential df of a function, either in the traditional language of infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

s or the modern language of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s.

• A differential of the form

is called a total differential or an exact differential
Exact differential
A mathematical differential is said to be exact, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q....

if it is the differential of a function. Again this can be interpreted infinitesimally, or by using differential forms and the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

.

• It is another name for the derivative as a linear map, i.e., if f is a differentiable function from Rn to Rm, then the (total) derivative (or differential) of f at x∈Rn is the linear map from Rn to Rm whose matrix is the Jacobian matrix of f at x.

• It is a synonym for the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

, which is essentially the derivative of a function from Rn to R.

• It is sometimes used as a synonym for the material derivative, , in fluid mechanics.

## 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 R2. In this case the partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

of f with respect to x does not give the true rate of change of f with respect to changing x because changing x necessarily changes y. The total derivative takes such dependencies into account.

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.
Notice that this is not equal to the partial derivative:.

While one can often perform substitutions to eliminate indirect dependencies, the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

provides for a more efficient and general technique. Suppose M(t, p1, ..., pn) is a function of time t and n variables which themselves depend on time. Then, the total time derivative of M is

The chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

for differentiating a function of several variables implies that

This expression is often used in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

for a gauge transformation of the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

, as two Lagrangians that differ only by the total time derivative of a function of time and the n generalized coordinates
Generalized coordinates
In the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration....

lead to the same equations of motion. 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 is a function of time t and n variables as in the previous section. Then, the differential of M is

This expression is often interpreted heuristically as a relation between infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

s. However, if the variables t and pj are interpreted as functions, and is interpreted to mean the composite of M with these functions, then the above expression makes perfect sense as an equality of differential 1-forms, and is immediate from the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

for the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

. The advantage of this point of view is that it takes into account arbitrary dependencies between the variables. For example, if then . In particular, if the variables pj are all functions of t, as in the previous section, then

## The total derivative as a linear map

Let be an open subset. Then a function is said to be (totally) differentiable at a point , if there exists a linear map (also denoted Dpf or Df(p)) such that

The linear map is called the (total) derivative or (total) differential of at . A function is (totally) differentiable if its total derivative exists at every point in its domain.

Note that f is differentiable if and only if each of its components is differentiable. For this it is necessary, but not sufficient, that the partial derivatives of each function fj exist. However, if these partial derivatives exist and are continuous, then f is differentiable and its differential at any point is the linear map determined by the Jacobian matrix of partial derivatives at that point.

## Total differential equation

A total differential equation is a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

expressed in terms of total derivatives. Since the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

is a natural operator, in a sense that can be given a technical meaning, such equations are intrinsic and geometric.

## Application of the total differential to error estimation

In measurement, the total differential is used in estimating the error
Experimental uncertainty analysis
The purpose of this introductory article is to discuss the experimental uncertainty analysis of a derived quantity, based on the uncertainties in the experimentally measured quantities that are used in some form of mathematical relationship to calculate that derived quantity...

Δf of a function f based on the errors Δx, Δy, ... of the parameters x, y, .... Assuming that the interval is short enough for the change to be approximately linear:
Δf(x) = f(x) × Δx

and that all variables are independent, then for all variables,

This is because the derivative fx with respect to the particular parameter x gives the sensitivity of the function f to a change in x, in particular the error Δx. As they are assumed to be independent, the analysis describes the worst-case scenario. The absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign. From this principle the error rules of summation, multiplication etc. are derived, e.g.:
Let f(a, b) = a × b;

Δf = faΔa + fbΔb; evaluating the derivatives

Δf = bΔa + aΔb; dividing by f, which is a × b

Δf/f = Δa/a + Δb/b

That is to say, in multiplication, the total relative error is the sum of the relative errors of the parameters.