Dt

Dt[f,x]

gives the total derivative df/dx.

Dt[f]

gives the total differential df.

Dt[f,{x,n}]

gives the multiple derivative dnf/dxn.

Dt[f,x1,x2,…]

gives d/dx1 d/dx2 … f.

Details and Options

  • Dt[f,x1,…,Constants->{c1,…}] specifies that the ci are constants, which have zero total derivative.
  • Symbols with attribute Constant are taken to be constants, with zero total derivative.
  • If an object is specified to be a constant, then all functions with that object as a head are also taken to be constants.
  • All quantities not explicitly specified as constants are assumed to depend on the xi.
  • You can specify total derivatives by assigning values to Dt[f], etc.

Examples

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Basic Examples  (4)

Find the total derivative with respect to :

Find the total differential of :

Find the second total derivative with respect to :

Find the total derivative with respect to two variables:

Scope  (4)

The total derivative of a constant is 0:

Total derivative of a general monomial:

Total derivative of a trigonometric function:

Rules for total differentiation including sums:

Products:

Quotients:

Composition:

Logarithmic total derivative:

Options  (3)

Constants  (3)

Specify that is constant with respect to :

Specify that and are constant with respect to :

Use TagSet to specify a constant:

Properties & Relations  (1)

The total derivative of a function of alone is the same as its derivative:

Wolfram Research (1988), Dt, Wolfram Language function, https://reference.wolfram.com/language/ref/Dt.html.

Text

Wolfram Research (1988), Dt, Wolfram Language function, https://reference.wolfram.com/language/ref/Dt.html.

CMS

Wolfram Language. 1988. "Dt." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Dt.html.

APA

Wolfram Language. (1988). Dt. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Dt.html

BibTeX

@misc{reference.wolfram_2024_dt, author="Wolfram Research", title="{Dt}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Dt.html}", note=[Accessed: 30-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_dt, organization={Wolfram Research}, title={Dt}, year={1988}, url={https://reference.wolfram.com/language/ref/Dt.html}, note=[Accessed: 30-December-2024 ]}