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:

Introduced in 1988
 (1.0)