# Total Derivatives

Dt[f] | total differential |

Dt[f,x] | total derivative |

Dt[f,x,y,...] | multiple total derivative |

Dt[f,x,Constants->{c_{1},c_{2},...}] | total derivative with constant (i.e. ) |

y/:Dt[y,x]=0 | set |

SetAttributes[c,Constant] | define c to be a constant in all cases |

Total differentiation operations.

When you find the derivative of some expression with respect to , you are effectively finding out how fast changes as you vary . Often will depend not only on , but also on other variables, say and . The results that you get then depend on how you assume that and vary as you change .

There are two common cases. Either and are assumed to stay fixed when changes, or they are allowed to vary with . In a standard *partial derivative* , all variables other than are assumed fixed. On the other hand, in the *total derivative* , all variables are allowed to change with .

In *Mathematica*, D[f, x] gives a partial derivative, with all other variables assumed independent of x. Dt[f, x] gives a total derivative, in which all variables are assumed to depend on x. In both cases, you can add an argument to give more information on dependencies.

This gives the

*partial derivative* .

is assumed to be independent of

.

Out[1]= | |

This gives the

*total derivative* . Now

is assumed to depend on

.

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You can make a replacement for

.

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You can also make an explicit definition for

. You need to use

to make sure that the definition is associated with

.

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With this definition made,

Dt treats

as independent of

.

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This removes your definition for the derivative of

.

This takes the total derivative, with

held fixed.

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This specifies that

is a constant under differentiation.

The variable

is taken as a constant.

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The

*function* is also assumed to be a constant.

Out[10]= | |

This gives the total differential

.

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You can make replacements and assignments for total differentials.

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