The term "total derivative" has several related meanings in mathematics. It is used as an alternate term for the convective
derivative .
It also denotes the derivative along a curve of a function depending on both a parameter
and intermediate variables . Let differentiable . The
chain rule gives
where the partial derivatives of partial
derivative
More generally, the total derivative of a function of several real variables denotes its Fréchet
derivative , the linear transformation
giving its first-order approximation. Its matrix in the
standard bases is the Jacobian
(Lebl 2026, pp. 35-40).
The total derivative of a function Wolfram
Language as Dt f [t ,
x , y , ...], t ].
See also Chain Rule ,
Convective Derivative ,
Derivative ,
Exact
Differential ,
Exterior Derivative ,
Fréchet Derivative ,
Jacobian ,
Partial Derivative
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References Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical
Encyclopaedia." Lebl,
J. Basic Analysis II: Introduction to Real Analysis, Vol. II, version 6.3.
pp. 35-41, 2026. https://www.jirka.org/ra/realanal2.pdf . Trott,
M. "Differentiation." §1.6.1 in The
Mathematica GuideBook for Symbolics. https://doi.org/10.1007/0-387-28815-5 .
https://www.mathematicaguidebooks.org/ . Referenced
on Wolfram|Alpha Total Derivative
Cite this as:
Weisstein, Eric W. "Total Derivative."
From MathWorld https://mathworld.wolfram.com/TotalDerivative.html
Subject classifications
, where 
and the 
are differentiable. The
chain rule gives
are evaluated at 
. The partial
derivative 
holds the 
fixed, whereas 
includes their dependence on 
(Lebl 2026, pp. 38-41).
with respect to 
is implemented in the Wolfram
Language as Dt[f[t,
x, y, ...], t].
