In mathematics, a total derivative may be either. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...

1. A synonym for the gradient of a function. See gradient for more information. In the case of a function of two variables the total derivative or gradient is the equation for the osculating plane.
   • The total derivative may also be used to refer to a differential one form which can be thought of as a generalization of the gradient to differential manifolds.
2. a derivative which takes indirect dependencies into account;

These two uses are usually readily distinguishable from context. A function f(x, y, z) several variables has only one gradient and one can thus talk about the total derivative of f. On the other hand when used in the second sense the total derivative must always be taken with respect to a particular variable, e.g., the total derivative of f(x, y,z) with respect to x. Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ... Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ... Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ... In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

Formal Definition

Let be open and be a map. f is said to be totally differentiable in , if there exists a linear map such that . The linear map D_pf is called the total derivative of f in p.

Totally Differentiable

A function is totally differentiable at a point if its total derivative or gradient exists at that point. The function is said to be totally differentiable if it is totally differentiable at every point in the function's domain. Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ...

Note that while the total derivative is a column vector consisting of the partial derivatives the existence of these partial derivatives is necessary but not sufficient for the total derivative to exist. 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 (as opposed to the total derivative, in which all variables are allowed to vary). ...

Intuitively this is because the total derivative at the point (a₀, a₁, ...) must become a sufficiently good approximation to the function as one approaches the point (a₀, a₁, ...) in any way. This approximation could be sufficiently good if you approach the function along any particular line allowing the partial derivatives to exist but not if approached in some more complex manner.

However, If the partial derivatives exist and are continuous on an open neighborhood surrounding the point then the total derivative is guaranteed to exist there.

Derivative taking indirect dependencies into account

The total derivative in this sense is simply a partial derivative of a function taking into account external dependencies, i.e., f is written as a function of x, y and z, but we take these variables to be dependent on each other instead of independent. For instance, we might wish to regard y and z as functions of x.

For example, suppose f (x, y, z) = xyz. Normally if you want to determine the rate of change of f with respect to x, you simply take the partial derivative of f with respect to x. In this case . However, if y and z are not truly independent but depend on x as well this technique is flawed. For instance if we set x = y = z then f(x) = xyz = x³ and . The total derivative of f with respect to x in this case would just be 3x².

While one can often perform substitutions to eliminate indirect dependencies, the chain rule provides for a more efficient and general technique. Suppose is a function of time t and n variables p_i which themselves depend on time. Then, the total time derivative of M is

The chain rule for differentiating a function of several variables implies that In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

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. Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ...

For example, the total derivative of f(x(t), y(t)) is

There is no term since f itself does not depend on an independent variable t directly.

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. An illustration of a differential equation. ...

Error estimation

In measurement^[1] , the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy, ... of the parameters x, y, .... Assuming that

Δf(x) = f'(x) × Δx

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

Δf = |f_x Δx |+ |f_y Δy| +...

This is because the derivative f_x 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 = f_aΔa + f_bΔb; 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. In the mathematical subfield of numerical analysis the approximation error in some data is the difference between the exact value and the value used. ...

Bibliography

• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2

• From thesaurus.maths.org total derivative

1. ^ http://www.tcd.ie/Economics/staff/obrienej/Calculus.pdf]