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In the [[mathematics|mathematical field]] of [[differential calculus]], the term '''total derivative''' has a number of closely related meanings.

<ul><li>The total derivative (full derivative) of a function <math>f</math>, of several variables, e.g., <math>t</math>, <math>x</math>, <math>y</math>, etc., with respect to one of its input variables, e.g., <math>t</math>, is different from its [[partial derivative|partial derivative (<math>\partial</math>)]]. Calculation of the total derivative of <math>f</math> with respect to <math>t</math> does not assume that the other arguments are constant while <math>t</math> varies; instead, it allows the other arguments to depend on <math>t</math>. The total derivative adds in these ''indirect dependencies'' to find the overall dependency of <math>f</math> on <math>t</math>. For example, the total derivative of <math>f(t,x,y)</math> with respect to <math>t</math> is

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

Which simplifies to

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

Consider multiplying both sides of the equation by the [[Differential (infinitesimal)|differential]] <math>\operatorname dt</math>:

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

The result will be the differential change <math>\operatorname df</math> in the function <math>f</math>. Because <math>f</math> depends on <math>t</math>, some of that change will be due to the [[partial derivative]] of <math>f</math> with respect to <math>t</math>. However, some of that change will also be due to the partial derivatives of <math>f</math> with respect to the variables <math>x</math> and <math>y</math>. So, the differential <math>\operatorname dt</math> is applied to the total derivatives of <math>x</math> and <math>y</math> to find differentials <math>\operatorname dx</math> and <math>\operatorname dy</math>, which can then be used to find the contribution to <math>\operatorname df</math>.</li>

It can also refer to differential operates that computes the total derivate.</ul>
* It refers to the (total) differential d''f'' of a function, either in the traditional language of [[infinitesimal]]s or the modern language of [[differential form]]s.

<ul><li>A differential of the form

:<math>\sum_{j=1}^k f_j(x_1,\dots, x_k) \operatorname d{x_j}</math>

is called a '''total differential''' or an '''[[exact differential]]''' if it is the differential of a function. Again this can be interpreted infinitesimally, or by using differential forms and the [[exterior derivative]].</li></ul>

* It is another name for the derivative as a linear map, i.e., if ''f'' is a [[differentiable function#Differentiability in higher dimensions|differentiable function]] from '''R'''''n'' to '''R'''''m'', then the (total) derivative (or differential) of ''f'' at ''x''∈'''R'''''n'' is the linear map from '''R'''''n'' to '''R'''''m'' whose matrix is the [[Jacobian matrix]] of ''f'' at ''x''.

* It is sometimes used as a synonym for the [[material derivative]], <math>\frac{D\mathbf{u}}{Dt}</math>, 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 ''R''2. In this case the [[partial derivative]] 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
:<math>f(x,y)=xy</math>.
The rate of change of ''f'' with respect to ''x'' is usually the partial derivative of ''f'' with respect to ''x''; in this case,
:<math>\frac{\partial f}{\partial x} = y</math>.
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
:<math>y=x;</math>
then
:<math>f(x,y) = f(x,x) = x^2</math>.
In that case, the total derivative of ''f'' with respect to ''x'' is
:<math>\frac{\mathrm{d}f}{\mathrm{d}x} = 2 x</math>.
Instead of immediately substituting for ''y'' in terms of ''x'', this can be found equivalently using the chain rule:

:<math>\frac{\mathrm{d}f}{\mathrm{d}x}= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}\frac{\mathrm{d}y}{\mathrm{d}x} = y+x \cdot 1 = x+y.</math>

Notice that this is not equal to the partial derivative:
:<math>\frac{\mathrm{d}f}{\mathrm{d}x} = 2 x \neq \frac{\partial f}{\partial x} = y = x</math>.

While one can often perform substitutions to eliminate indirect dependencies, the [[chain rule]] provides for a more efficient and general technique. Suppose ''M''(''t'', ''p''1, ..., ''pn'') is a function of time ''t'' and ''n'' variables <math>p_i</math> which themselves depend on time. Then, the total time derivative of ''M'' is

:<math> {\operatorname{d}M \over \operatorname{d}t} = \frac{\operatorname d}{\operatorname d t} M \bigl(t, p_1(t), \ldots, p_n(t)\bigr).</math>

The [[chain rule]] for differentiating a function of several variables implies that

:<math> {\operatorname{d}M \over \operatorname{d}t}
= \frac{\partial M}{\partial t} + \sum_{i=1}^n \frac{\partial M}{\partial p_i}\frac{\operatorname{d}p_i}{\operatorname{d}t}
= \biggl(\frac{\partial}{\partial t} + \sum_{i=1}^n \frac{\operatorname{d}p_i}{\operatorname{d}t}\frac{\partial}{\partial p_i}\biggr)(M).</math>

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 absorber theory#Resolution_of_causality_issue|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

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

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 <math>M(t,p_1,\dots,p_n)</math> is a function of time ''t'' and ''n'' variables <math>p_i</math> as in the previous section. Then, the differential of ''M'' is

:<math> \operatorname d M = \frac{\partial M}{\partial t} \operatorname d t + \sum_{i=1}^n \frac{\partial M}{\partial p_i}\operatorname{d}p_i.</math>

This expression is often interpreted ''heuristically'' as a relation between [[infinitesimal]]s. However, if the variables ''t'' and ''p''''j'' are interpreted as functions, and <math>M(t,p_1,\dots,p_n)</math> is interpreted to mean the composite of ''M'' with these functions, then the above expression makes perfect sense as an equality of [[differential 1-form]]s, and is immediate from the [[chain rule]] for the [[exterior derivative]]. The advantage of this point of view is that it takes into account arbitrary dependencies between the variables. For example, if <math>p_1^2=p_2 p_3</math> then <math>2p_1\operatorname dp_1=p_3 \operatorname d p_2+p_2\operatorname d p_3</math>. In particular, if the variables ''p''''j'' are all functions of ''t'', as in the previous section, then

:<math> \operatorname d M
= \frac{\partial M}{\partial t} \operatorname d t + \sum_{i=1}^n \frac{\partial M}{\partial p_i}\frac{\partial p_i}{\partial t}\,\operatorname d t.</math>

==The total derivative as a linear map==

Let <math>U\subseteq \mathbb{R}^{n}</math> be an [[open subset]]. Then a function <math>f:U\rightarrow \mathbb{R}^m</math> is said to be ('''totally''') '''differentiable''' at a point <math>p\in U</math>, if there exists a linear map <math>\operatorname df_p:\mathbb{R}^n \rightarrow \mathbb{R}^m</math> (also denoted D''p''''f'' or D''f''(p)) such that

:<math>\lim_{x\rightarrow p}\frac{\|f(x)-f(p)-\operatorname df_p(x-p)\|}{\|x-p\|}=0.</math>

The linear map <math>\operatorname d f_p</math> is called the ('''total''') '''derivative''' or ('''total''') '''differential''' of <math>f</math> at <math>p</math>. 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 <math>f_i:U\rightarrow \mathbb{R}</math> is differentiable. For this it is necessary, but not sufficient, that the partial derivatives of each function ''f''''j'' 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==
{{main|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 of the total differential to error estimation ==

In measurement, the total differential is used in [[Experimental uncertainty analysis|estimating the error]] Δ''f'' of a function ''f'' based on the errors Δ''x'', Δ''y'', ... of the parameters ''x, y, ...''. Assuming that the interval is short enough for the change to be approximately linear:

:Δ''f''(''x'') = ''f'''(''x'') × Δ''x''

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

:<math>\Delta f = f_x \Delta x + f_y \Delta y + \cdots</math>

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 Δ''x''. 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'';

:Δ''f'' = ''f''''a''Δ''a'' + ''f''''b''Δ''b''; evaluating the derivatives

:Δ''f'' = ''b''Δ''a'' + ''a''Δ''b''; dividing by ''f'', which is ''a'' × ''b''

:Δ''f''/''f'' = Δ''a''/''a'' + Δ''b''/''b''

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

To illustrate how this depends on the function considered, consider the case where the function is ''f(a, b) = a ln b'' instead. Then, it can be computed that the error estimate is
:Δ''f''/''f'' = Δ''a''/''a'' + Δ''b''/(''b'' ln ''b'')
with an extra 'ln ''b''' factor not found in the case of a simple product. This additional factor tends to make the error smaller, as ln ''b'' is not as large as a bare ''b''.

== References ==

* 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 [http://thesaurus.maths.org/mmkb/entry.html;jsessionid=EC2A4288632FF1D59B1207BA04FCC65B?action=entryByConcept&id=952&langcode=en total derivative]

== External links ==
*{{MathWorld|TotalDerivative|Total Derivative}}
* http://www.sv.vt.edu/classes/ESM4714/methods/df2D.html

[[Category:Multivariable calculus]]
[[Category:Differential calculus]]
[[Category:Differential operators]]
[[Category:Lagrangian mechanics]]
[[Category:Mathematical analysis]]

[[ja:偏微分#全微分]]

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