# Total derivative

The **total derivative** is the derivative of a function of several variables taken as a whole: a linear map that gives the best linear approximation to the function's change near a point, accounting for movement in all variables at once. In multivariable real analysis it is the linear map \( L : V \to W \) satisfying \( f(a+\Delta) - f(a) - L(\Delta) = o(\lVert \Delta \rVert) \) as \( \Delta \to 0 \), and in fluid mechanics and thermodynamics the same phrase refers to related operators built from partial derivatives and the chain rule.<sup>[1](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition (linear map form) | For \( f : U \to W \) on an open subset of finite-dimensional real vector spaces, the total derivative at \( a \) is the linear map \( L \) with \( \lim_{\Delta \to 0} \frac{f(a+\Delta) - f(a) - L(\Delta)}{\lVert \Delta \rVert} = 0 \); it is denoted \( Df_a \)<sup>[1](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)</sup> |
| Matrix form | When \( f : \mathbb{R}^n \to \mathbb{R}^m \) is partially differentiable at \( \xi \) and the Jacobian matrix \( J_f(\xi) \) satisfies the same limit condition, \( J_f(\xi) \) is called the total derivative, written \( Df(\xi) \); this definition implies continuity<sup>[2](https://link.springer.com/article/10.1007/s11858-024-01584-w)</sup> |
| Relation to directional derivatives | \( Df_a(v) \) equals the directional derivative of \( f \) at \( a \) in the direction \( v \); the total derivative can fail to exist at a point even when all directional derivatives exist<sup>[1](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)</sup> |
| Fluid mechanics form | The material (total, substantial, convective) derivative is \( \frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla \), the time rate of change following a moving fluid particle<sup>[3](https://www.continuummechanics.org/materialderivative.html)</sup> |
| Total differential | For \( z = f(x,y) \), \( dz = f_x(x,y)\,dx + f_y(x,y)\,dy \); historically the total differential is as old as the calculus itself, while partial derivatives were defined only in the 18th century<sup>[4](https://scholarlyworks.adelphi.edu/esploro/outputs/journalArticle/From-Differentials-to-Derivatives/991004547681806266)</sup> |
| Chain rule | For \( z = f(x,y) \) with \( x = g(t) \), \( y = h(t) \), the total derivative along the path is \( \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt} \)<sup>[5](https://lemesurierb.people.charleston.edu/math221-notes-and-study-guide/section_thechainruleIFT.html)</sup> |

## Definition as a linear map

The modern definition treats the derivative of a multivariable function as a linear transformation rather than a number. Let \( V \) and \( W \) be finite-dimensional real vector spaces, \( U \) an open subset of \( V \), and \( f : U \to W \). The total derivative of \( f \) at \( a \), when it exists, is the linear map \( L : V \to W \) such that

\[ \lim_{\Delta \to 0} \frac{f(a+\Delta) - f(a) - L(\Delta)}{\lVert \Delta \rVert} = 0, \]

that is, \( f(a+\Delta) - f(a) - L(\Delta) = o(\lVert \Delta \rVert) \). On a finite-dimensional space this property does not depend on the norm chosen. The derivative is denoted \( Df_a \), so that \( f(a+\Delta) = f(a) + Df_a(\Delta) + o(\lVert \Delta \rVert) \). Some texts call this object the *differential* of \( f \) at \( a \) and write \( df_a \) or \( df(a) \).<sup>[1](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)</sup>

Conceptually, \( Df_a \) is the best linear approximation to \( f(a+h) - f(a) \) for small \( h \).<sup>[6](https://maths.free/multivariable-calculus/total-derivative/)</sup>

## Matrix form and the Jacobian

Many textbooks instead give a constructive definition through partial derivatives. A function \( f : \mathbb{R}^n \to \mathbb{R}^m \) is totally differentiable at \( \xi \) if it is partially differentiable there and the Jacobian matrix \( J_f(\xi) \in \mathbb{R}^{m \times n} \), which collects all the partial derivatives, satisfies

\[ \lim_{h \to 0} \frac{f(\xi+h) - f(\xi) - J_f(\xi) \cdot h}{\lVert h \rVert} = 0. \]

In that case \( J_f(\xi) \) is called the total derivative of \( f \) at \( \xi \), written \( Df(\xi) \).<sup>[2](https://link.springer.com/article/10.1007/s11858-024-01584-w)</sup> The extra limit condition is needed because the mere existence of all partial derivatives does not guarantee that the matrix they form approximates the function locally; the constructive definition, by requiring the limit, implies continuity of \( f \) at the point.<sup>[2](https://link.springer.com/article/10.1007/s11858-024-01584-w)</sup>

The two formulations describe the same object. Choosing bases for \( V \) and \( W \), the matrix of \( Df_a \) has entries given by the partial derivatives, and for a differentiable \( f : \mathbb{R}^n \to \mathbb{R}^m \) the derivative matrix is the Jacobian matrix.<sup>[1](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)</sup><sup> • </sup><sup>[7](https://www2.math.upenn.edu/~gressman/analysis/12-totalderivs.html)</sup>

## Relation to directional and partial derivatives

The total derivative determines every directional derivative. For \( v \neq 0 \), substituting \( \Delta = t \cdot v \) into the definition shows that \( Df_a(v) \) equals the directional derivative of \( f \) at \( a \) in the direction \( v \), that is \( \lim_{t \to 0} \frac{f(a+tv) - f(a)}{t} \).<sup>[1](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)</sup> The converse fails: there are examples where the total derivative fails to exist at a point even though all directional derivatives exist, because existence of derivatives along individual directions does not control the function's behavior across all directions simultaneously.<sup>[1](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)</sup>

## The chain rule and total derivatives along paths

When the variables of a function depend on a common parameter, the chain rule gives the total rate of change along that dependence. For \( z = f(x,y) \) with \( x = g(t) \) and \( y = h(t) \),

\[ \frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}. \]

Here \( x \) and \( y \) are called intermediate variables, to distinguish them from the independent variable \( t \).<sup>[5](https://lemesurierb.people.charleston.edu/math221-notes-and-study-guide/section_thechainruleIFT.html)</sup> In the general version, for \( z = f(x_1, \dots, x_n) \) with \( x_i = g_i(t_1, \dots, t_m) \), each partial derivative \( \partial z / \partial t_i \) is the sum \( \sum_{j=1}^{n} \frac{\partial z}{\partial x_j} \frac{\partial x_j}{\partial t_i} \).<sup>[5](https://lemesurierb.people.charleston.edu/math221-notes-and-study-guide/section_thechainruleIFT.html)</sup> This is the form used when a quantity depends on position and time and the position itself moves.

## The material derivative in fluid mechanics

In continuum mechanics the phrase "total derivative" names an operator, not a matrix. The material derivative computes the time rate of change of a quantity such as temperature or velocity for a portion of material moving with velocity \( \mathbf{v} \); for a fluid, the movement is the flow field itself. It is also called the convective, substantial, or substantive derivative, and is usually written \( D/Dt \) or \( d/dt \):<sup>[3](https://www.continuummechanics.org/materialderivative.html)</sup>

\[ \frac{D}{Dt} = \frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla. \]

The operator translates the Lagrangian time derivative, which follows a particle, into Eulerian field language, in which coordinates are fixed in space and fluid streams through them.<sup>[3](https://www.continuummechanics.org/materialderivative.html)</sup><sup> • </sup><sup>[8](https://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec02/lecture2.html)</sup> For a temperature field \( T(t,x,y,z) \), applying the chain rule gives \( \frac{dT}{dt} = \frac{\partial T}{\partial t} + v_x \frac{\partial T}{\partial x} + v_y \frac{\partial T}{\partial y} + v_z \frac{\partial T}{\partial z} = \frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T \).<sup>[3](https://www.continuummechanics.org/materialderivative.html)</sup> The two terms have standard names: in the acceleration \( \frac{D\mathbf{v}}{Dt} = \frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \), the first term is the local acceleration and the second the convective acceleration.<sup>[8](https://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec02/lecture2.html)</sup> A steady flow can still have nonzero acceleration, because a particle speeding up through a converging nozzle registers no change at a fixed point in space; the material derivative corrects for exactly this effect.<sup>[3](https://www.continuummechanics.org/materialderivative.html)</sup>

## The total differential in thermodynamics

In thermodynamics the related object is the total differential. For a function \( u(x_1, x_2, \dots, x_n) \),

\[ du = \left( \frac{\partial u}{\partial x_1} \right) dx_1 + \left( \frac{\partial u}{\partial x_2} \right) dx_2 + \cdots + \left( \frac{\partial u}{\partial x_n} \right) dx_n, \]

the sum of all partial derivatives, each multiplied by an infinitesimal change along its coordinate.<sup>[9](https://chem.libretexts.org/Courses/University_of_Wisconsin_Oshkosh/Chem_371%3A_P-Chem_2_to_Folow_Combined_Biophysical_and_P-Chem_1_%28Gutow%29/01%3A_Thermodynamics/1.01%3A_Review_and_Setting_the_Stage)</sup> For the internal energy \( U(V,T) \) of a closed system this reads \( dU = \left( \frac{\partial U}{\partial V} \right)_T dV + \left( \frac{\partial U}{\partial T} \right)_V dT \).<sup>[9](https://chem.libretexts.org/Courses/University_of_Wisconsin_Oshkosh/Chem_371%3A_P-Chem_2_to_Folow_Combined_Biophysical_and_P-Chem_1_%28Gutow%29/01%3A_Thermodynamics/1.01%3A_Review_and_Setting_the_Stage)</sup> The form is useful when the partial derivatives can be measured experimentally but the underlying function is unknown; for a fluid volume \( V(n,T,P) \), the tabulated coefficient of thermal expansion \( \alpha \), isothermal compressibility \( \kappa \), and molar volume \( V_m \) turn the total differential into \( dV = \alpha V\,dT - \kappa V\,dP + V_m\,dn \).<sup>[10](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Chemical_Thermodynamics_%28Supplement_to_Shepherd_et_al.%29/09%3A_Fundamental_7_-_Variable_Changes/9.03%3A_The_Total_Differential)</sup>

## History and generalization

The historical order of the ideas is the reverse of the textbook order. The total differential \( dz = f_x\,dx + f_y\,dy \) is as old as the calculus itself, appearing in the work of the 17th century, whereas partial derivatives were only defined in the 18th century.<sup>[4](https://scholarlyworks.adelphi.edu/esploro/outputs/journalArticle/From-Differentials-to-Derivatives/991004547681806266)</sup>

Defining the total derivative as a linear mapping rather than a matrix, a change from the one-variable case where \( f'(\xi) \) is a real number, has two advantages. The derivative itself is then the linear map used for the local approximation \( f(\xi+h) - f(\xi) \approx A_{\xi}(h) \), and the tangent plane can be defined as the graph of that map. In addition, a linear mapping can be specified without choosing a basis, which allows the same definition for functions \( V \to W \) between arbitrary Banach spaces, finite or infinite dimensional.<sup>[2](https://link.springer.com/article/10.1007/s11858-024-01584-w)</sup>

## References

1. [Notes of lectures on Multivariable Calculus, University of Oxford](https://people.maths.ox.ac.uk/~tillmann/background-multvar.pdf)
2. [The meaning landscape of the concept of the total derivative in multivariable real analysis textbooks, ZDM – Mathematics Education, Springer](https://link.springer.com/article/10.1007/s11858-024-01584-w)
3. [Material Derivative, continuummechanics.org](https://www.continuummechanics.org/materialderivative.html)
4. [Robert E. Bradley and Salvatore J. Petrilli, From Differentials to Derivatives, Notices of the American Mathematical Society 72(6), 624–635 (2025)](https://scholarlyworks.adelphi.edu/esploro/outputs/journalArticle/From-Differentials-to-Derivatives/991004547681806266)
5. [The Chain Rule and Implicit Differentiation, College of Charleston lecture notes](https://lemesurierb.people.charleston.edu/math221-notes-and-study-guide/section_thechainruleIFT.html)
6. [Total derivative, step-by-step with animation, maths.free](https://maths.free/multivariable-calculus/total-derivative/)
7. [Advanced Analysis: Total Derivatives, University of Pennsylvania](https://www2.math.upenn.edu/~gressman/analysis/12-totalderivs.html)
8. [Material/Substantial Derivative, MIT fluids module](https://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec02/lecture2.html)
9. [Total derivatives, Chemistry LibreTexts](https://chem.libretexts.org/Courses/University_of_Wisconsin_Oshkosh/Chem_371%3A_P-Chem_2_to_Folow_Combined_Biophysical_and_P-Chem_1_%28Gutow%29/01%3A_Thermodynamics/1.01%3A_Review_and_Setting_the_Stage)
10. [The Total Differential, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Chemical_Thermodynamics_%28Supplement_to_Shepherd_et_al.%29/09%3A_Fundamental_7_-_Variable_Changes/9.03%3A_The_Total_Differential)

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